Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...
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Indeterminate Forms L’Hopital’s Rule Examples
Lecture 19Section 10.5 Indeterminate Form (0/0)
Section 10.6 Other Indeterminate Forms (∞/∞), (0 · ∞),· · ·
Jiwen He
Department of Mathematics, University of Houston
[email protected]://math.uh.edu/∼jiwenhe/Math1432
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 1 / 15
![Page 2: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/2.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 3: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/3.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 4: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/4.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 5: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/5.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 6: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/6.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 7: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/7.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 8: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/8.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 9: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/9.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 10: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/10.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 11: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/11.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 12: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/12.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 13: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/13.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 14: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/14.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 15: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/15.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 16: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/16.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 17: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/17.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
What is the Indeterminate Form (0/0)?Example
limx→2
x − 2
x2 − 4= lim
x→2
x − 2
(x − 2)(x + 2)= lim
x→2
1
x + 2=
1
2 + 2=
1
4
(L’Hopital’s Rule) = limx→2
f ′(x)
g ′(x)= lim
x→2
1
2x=
1
2 · 2=
1
4(Amazing!!)
As x → 2, f (x) = x − 2 → 0 and g(x) = x2 − 4 → 0, but thequotient
f (x)
g(x)=
x − 2
x2 − 4→ 0
0.
We can not determine a value for “00” without some extra work!
If limx→a
f (x) = limx→a
g(x) = 0, then limx→a
f (x)
g(x)might equal any
number or even fail to exist!
Condensed: The “form” 00 is indeterminate.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 2 / 15
![Page 18: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/18.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 19: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/19.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 20: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/20.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 21: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/21.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 22: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/22.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 23: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/23.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 24: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/24.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 25: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/25.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 26: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/26.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 27: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/27.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 28: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/28.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 29: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/29.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Indeterminate FormsIndeterminate Forms
The most basic indeterminate form is 00 .
It is indeterminate because, if limx→a
f (x) = limx→a
g(x) = 0, then
limx→a
f (x)
g(x)might equal any number or even fail to exist!
Specific cases: limx→0
sin x
x, lim
x→∞
e−x
sin(1/x), · · · .
The forms ∞∞ and 0 · ∞ are also indeterminate.
They are actually equivalent to 00 , since any specific case of
one can be recast as a specific case of another.
For example,
limx→0+
x ln x = limx→0+
ln x
1/x= lim
x→0+
x
1/ ln x
0 · ∞ ∞∞
0
0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 3 / 15
![Page 30: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/30.jpg)
Indeterminate Forms L’Hopital’s Rule Examples (0/0) (∞/∞) and Others
Quiz
Quiz
1. Give the limit for
{1− n
n + 2
}∞n=1
(a) 0, (b) 1, (c) 2.
2. Give the limit for
{1 +
n
n + 2
}∞n=1
(a) 0, (b) 1, (c) 2.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 4 / 15
![Page 31: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/31.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 32: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/32.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 33: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/33.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 34: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/34.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 35: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/35.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 36: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/36.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 37: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/37.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 38: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/38.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 39: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/39.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 40: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/40.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 41: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/41.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 42: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/42.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 43: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/43.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 44: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/44.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 45: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/45.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 46: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/46.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 47: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/47.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 48: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/48.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 49: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/49.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 50: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/50.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
L’Hopital’s Rule
L’Hopital’s Rule
Suppose that limx→?
f (x)
g(x)is a
0
0or∞∞
. Then limx→?
f (x)
g(x)= lim
x→?
f ′(x)
g ′(x)
Examples
limx→2
x − 2
x2 − 4
(0
0
)= lim
x→2
1
2x=
1
2 · 2=
1
4
limx→0
sin x
x
(0
0
)= lim
x→0
cos x
1=
1
1= 1
limx→0
x − sin x
x sin x
(0
0
)= lim
x→0
1− cos x
sin x + x cos x
(0
0
)still
= limx→0
sin x
cos x + cos x − x sin x=
0
2= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 5 / 15
![Page 51: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/51.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 52: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/52.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 53: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/53.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 54: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/54.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 55: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/55.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 56: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/56.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 57: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/57.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 58: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/58.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 59: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/59.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 60: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/60.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
What is Wrong with This?
What is wrong with the following argument?
limx→0+
x2
sin x
(0
0
)= lim
x→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Remark
L’Hopital’s rule does not apply in cases where the numerator or thedenominator has a finite non-zero limit!!!For example,
limx→0+
2x
cos x=
0
1= 0,
but a blind application of L’Hopital’s rule leads incorrectly to
limx→0+
2x
cos x= lim
x→0+
2
− sin x=
2
−0= −∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 6 / 15
![Page 61: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/61.jpg)
Indeterminate Forms L’Hopital’s Rule Examples Rule
Quiz
Quiz
3. Give the limit for{
n1n
}∞n=1
(a) 1, (b) 2, (c) e.
4. Give the limit for
{(1 +
1
n
)n}∞n=1
(a) 1, (b) 2, (c) e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 7 / 15
![Page 62: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/62.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 63: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/63.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 64: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/64.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 65: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/65.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 66: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/66.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 67: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/67.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 68: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/68.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 69: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/69.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 70: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/70.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 71: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/71.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 72: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/72.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 73: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/73.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 74: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/74.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 75: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/75.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 76: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/76.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 77: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/77.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 78: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/78.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 79: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/79.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 80: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/80.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 81: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/81.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form 00
Examples
limx→0
sin x − x
x3
(0
0
)= lim
x→0
cos x − 1
3x2
(0
0
)still!
= limx→0
− sin x
6x
(0
0
)still! = lim
x→0
− cos x
6=−1
6
limx→0
2 cos x − 2 + x2
x4
(0
0
)= lim
x→0
−2 sin x + 2x
4x3
(0
0
)still!
= limx→0
−2 cos x + 2
12x2
(0
0
)still!
= limx→0
2 sin x
24x
(0
0
)still! = lim
x→0
2 cos x
24=
2
24=
1
12
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 8 / 15
![Page 82: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/82.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 83: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/83.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 84: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/84.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 85: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/85.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 86: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/86.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 87: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/87.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 88: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/88.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 89: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/89.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 90: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/90.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 91: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/91.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 92: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/92.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 93: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/93.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 94: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/94.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 95: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/95.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞∞
Examples
limx→∞
x2
ex
(∞∞
)= lim
x→∞
2x
ex
(∞∞
)still!
= limx→∞
2
ex=
2
∞= 0
limx→∞
ln(3x + 2x)
x
(∞∞
)= lim
x→∞
3x ln 2 + 2x ln 2
3x + 2x
= limx→∞
ln 3 + (2/3)x ln 2
1 + (2/3)x=
ln 3 + 0
1 + 0= ln 3
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 9 / 15
![Page 96: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/96.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 97: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/97.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 98: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/98.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 99: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/99.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 100: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/100.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 101: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/101.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 102: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/102.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 103: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/103.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 104: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/104.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 105: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/105.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 106: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/106.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 107: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/107.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 108: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/108.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 109: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/109.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 110: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/110.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Proof.
limx→∞
ln x
xα
(∞∞
)= lim
x→∞
1/x
αxα−1= lim
x→∞
1
αxα−1= 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 111: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/111.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Remark
ln x tends to infinity slower than any positive power of x .
Any positive power of ln x tends to infinity slower than x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 112: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/112.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ln x
limx→0+
xα ln x = 0, α > 0.
Proof.
limx→0+
xα ln x (0 · ∞) = limx→0+
ln x
x−α
(∞∞
)= lim
x→0+
1/x
−αx−α−1= lim
x→0+
xα
−α= 0
limx→∞
ln x
xα= 0, α > 0.
Remark
ln x tends to infinity slower than any positive power of x .
Any positive power of ln x tends to infinity slower than x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 10 / 15
![Page 113: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/113.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 114: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/114.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 115: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/115.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 116: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/116.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 117: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/117.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 118: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/118.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 119: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/119.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 120: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/120.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 121: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/121.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 122: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/122.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 123: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/123.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 124: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/124.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Limit Properties of ex
limx→∞
xn
ex= 0.
Proof.
limx→∞
xn
ex
(∞∞
)= lim
x→∞
nxn−1
ex
(∞∞
)= lim
x→∞
n(n − 1)xn−2
ex
(∞∞
)= · · ·
= limx→∞
n!
ex=
n!
∞= 0
Remark
ex tends to infinity fasterr than any positive power of x .
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 11 / 15
![Page 125: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/125.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 126: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/126.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 127: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/127.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 128: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/128.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 129: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/129.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 130: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/130.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 131: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/131.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 132: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/132.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 133: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/133.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 134: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/134.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 135: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/135.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 136: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/136.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 137: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/137.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 138: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/138.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 139: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/139.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 140: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/140.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 141: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/141.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 142: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/142.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Exponential Forms: 1∞, 00, and ∞0
Since ln xp = p ln x , the log can be used to convert each ofexponential forms 1∞, 00, and ∞0 to 0 · ∞:
ln(1∞) = ∞ · 0, ln(00) = ∞ · 0, ln(∞0) = 0 · ∞.
If limx→?
ln f (x) = L, then limx→?
f (x) = eL.
Example
limx→0+
xx(00
)= e0 = 1
limx→0+
ln(xx)(ln 00
)= lim
x→0+x ln x (0 · ∞)
= limx→0+
ln x
1/x
(∞∞
)= lim
x→0+
1/x
−1/x2= lim
x→0+(−x) = 0
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 12 / 15
![Page 143: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/143.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 144: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/144.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 145: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/145.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 146: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/146.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 147: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/147.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 148: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/148.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 149: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/149.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 150: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/150.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 151: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/151.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 152: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/152.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 153: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/153.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 154: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/154.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 155: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/155.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 156: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/156.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 157: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/157.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 158: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/158.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 159: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/159.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 160: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/160.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 161: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/161.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 162: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/162.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 163: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/163.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
More Exponential Forms: 1∞, 00, and ∞0
Examples
limx→∞
x1/x(∞0
)= e0 = 1
limx→0+
(1 + x)1/x (1∞) = e1 = e
limx→∞
ln x1/x(ln∞0
)= lim
x→∞
ln x
x
(∞∞
)= lim
x→∞
1/x
1= 0
limx→0+
ln(1 + x)1/x (ln 1∞) = limx→0+
ln(1 + x)
x
(0
0
)= lim
x→0+
1
1 + x= 1
Set x = n, we have the familiar result: as n →∞, n1n → 1.
Set x = 1/n, we have: as n →∞,(1 + 1
n
)n → e.
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 13 / 15
![Page 164: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/164.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 165: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/165.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 166: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/166.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 167: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/167.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 168: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/168.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 169: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/169.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 170: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/170.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 171: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/171.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 172: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/172.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 173: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/173.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 174: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/174.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 175: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/175.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 176: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/176.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 177: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/177.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 178: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/178.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 179: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/179.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 180: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/180.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 181: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/181.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 182: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/182.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 183: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/183.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Form ∞−∞The form ∞−∞ can be handled by converting it to a quotient.
limx→(π/2)−
(tan x − sec x) (∞−∞) = limx→(π/2)−
sin x − 1
cos x
(0
0
)= lim
x→(π/2)−
cos x
− sin x=
0
−1= 0
limx→∞
(ex+e−x − ex
)(∞−∞) = lim
x→∞
ee−x − 1
e−x
(0
0
)= lim
x→∞
ee−x(−e−x)
−e−x= lim
x→∞ee−x
= e0 = 1
tan x − sec x =sin x
cos x− 1
cos x=
sin x − 1
cos x
ex+e−x − ex = ex(ee−x − 1
)=
ee−x − 1
e−x
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 14 / 15
![Page 184: Lecture 19 - Section 10.5 Indeterminate Form (0/0) Section ...](https://reader030.fdocuments.net/reader030/viewer/2022012418/6172bf08d0652333fa032a5a/html5/thumbnails/184.jpg)
Indeterminate Forms L’Hopital’s Rule Examples 00∞∞ ln x and ex Exponential Forms ∞−∞
Outline
Indeterminate FormsIndeterminate Form (0/0)Other Indeterminate Forms
L’Hopital’s RuleL’Hopital’s Rule
More ExamplesForm 0
0Form ∞
∞Limit Properties of ln x and ex
Exponential Forms: 1∞, 00, and ∞0
Form ∞−∞
Jiwen He, University of Houston Math 1432 – Section 26626, Lecture 19 March 25, 2008 15 / 15