Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial...
Transcript of Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial...
![Page 1: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/1.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 2: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/2.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.
x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 3: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/3.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x
2.5937 2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 4: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/4.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937
2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 5: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/5.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481
2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 6: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/6.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692
2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 7: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/7.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814
2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 8: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/8.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814 2.71826
→ e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 9: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/9.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·
This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 10: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/10.jpg)
Special Limitsdefinition of e
The number e is defined as a limit. Here is one definition:
e = limx→0+
(1 + x)1x
A good way to evaluate this limit is make a table of numbers.x .1 .01 0.001 0.0001 0.00001 → 0
(1 + x)1x 2.5937 2.70481 2.71692 2.71814 2.71826 → e
Where e = 2.7 1828 1828 · · ·This limit will give the same result:
e = limx→∞
(1 +
1
x
)x
![Page 11: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/11.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 12: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/12.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x
2.70481 2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 13: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/13.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481
2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 14: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/14.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692
2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 15: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/15.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815
2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 16: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/16.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815 2.71827
→ e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 17: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/17.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 18: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/18.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 19: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/19.jpg)
Special Limitsalternate definition of e
Here is an equivalient definition for e:
e = limx→∞
(1 +
1
x
)x
x 100 1000 10000 1000000 →∞(1 + x)
1x 2.70481 2.71692 2.71815 2.71827 → e
Note thatlimx→0
f(x)
is the same as
limx→∞
f
(1
x
)
![Page 20: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/20.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 21: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/21.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·
I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 22: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/22.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 23: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/23.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 24: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/24.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 25: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/25.jpg)
Special Limitse the natural base
I the number e is the natural base in calculus. Manyexpressions in calculus are simpler in base e than in otherbases like base 2 or base 10
I e = 2.71828182845904509080 · · ·I e is a number between 2 and 3. A little closer to 3.
I e is easy to remember to 9 decimal places because 1828repeats twice: e = 2.718281828. For this reason, do not use2.7 to extimate e.
I use the ex button on your calculator to find e. Use 1 for x.
I example: F = Pert is often used for calculating compoundinterest in business applications.
![Page 26: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/26.jpg)
Special Limitsf(x) = 1
x
Here are four useful limits:
I
limx→+∞
1
x→ 0+ = 0
I
limx→−∞
1
x→ 0− = 0
I
limx→0+
1
x→ +∞
I
limx→0−
1
x→ −∞
![Page 27: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/27.jpg)
Special Limitsf(x) = 1
x
Here are four useful limits:
I
limx→+∞
1
x→ 0+ = 0
I
limx→−∞
1
x→ 0− = 0
I
limx→0+
1
x→ +∞
I
limx→0−
1
x→ −∞
![Page 28: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/28.jpg)
Special Limitsf(x) = 1
x
Here are four useful limits:
I
limx→+∞
1
x→ 0+ = 0
I
limx→−∞
1
x→ 0− = 0
I
limx→0+
1
x→ +∞
I
limx→0−
1
x→ −∞
![Page 29: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/29.jpg)
Special Limitsf(x) = 1
x
Here are four useful limits:
I
limx→+∞
1
x→ 0+ = 0
I
limx→−∞
1
x→ 0− = 0
I
limx→0+
1
x→ +∞
I
limx→0−
1
x→ −∞
![Page 30: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/30.jpg)
Special Limitsf(x) = 1
x cont.
The general idea is this:
I1
+BIG→ +small
I1
−BIG→ −small
I1
+small→ +BIG
I1
−small→ −BIG
![Page 31: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/31.jpg)
Special Limitsf(x) = 1
x cont.
The general idea is this:
I1
+BIG→ +small
I1
−BIG→ −small
I1
+small→ +BIG
I1
−small→ −BIG
![Page 32: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/32.jpg)
Special Limitsf(x) = 1
x cont.
The general idea is this:
I1
+BIG→ +small
I1
−BIG→ −small
I1
+small→ +BIG
I1
−small→ −BIG
![Page 33: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/33.jpg)
Special Limitsf(x) = 1
x cont.
The general idea is this:
I1
+BIG→ +small
I1
−BIG→ −small
I1
+small→ +BIG
I1
−small→ −BIG
![Page 34: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/34.jpg)
Special Limitsf(x) = 1
x
x
f
+1.0 +2.5 +4.0 +5.5 +7.0 +8.5−1.0−2.5−4.0−5.5−7.0−8.5
−9.0
−8.0
−7.0
−6.0
−5.0
−4.0
−3.0
−2.0
−1.0
+1.0
+2.0
+3.0
+4.0
+5.0
+6.0
+7.0
+8.0
+9.0
![Page 35: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/35.jpg)
Special Limits1
BIG = small, 1small = BIG
Look at 1x using some numbers:
x 10 100 1000 10000 100000 → +∞f(x) = 1
x .1 .01 .001 .0001 .00001 → 0+
x -10 -100 -1000 -10000 -100000 → −∞f(x) = 1
x -.1 -.01 -.001 -.0001 -.00001 → −0+
x .1 .01 .001 .0001 .00001 → +0
f(x) = 1x 10 100 1000 10000 100000 → +∞
x -.1 -.01 -.001 -.0001 -.00001 → −0
f(x) = 1x -10 -100 -1000 -10000 -100000 → −∞
![Page 36: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/36.jpg)
Special Limitsx→ ±∞ for Polynomials
I when taking limits of polynomials to ±∞ drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.
I this is an intermediate step in taking the limit. Use algebra tosimplify the expression at this step then continue to work onfinding the limit to infinity.
I this works because for large values of x the highest powerterm of the polynomial is so much larger that all of thesmaller degree terms that the smaller degree terms have noeffect in the limit to infinity.
![Page 37: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/37.jpg)
Special Limitsx→ ±∞ for Polynomials
I when taking limits of polynomials to ±∞ drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.
I this is an intermediate step in taking the limit. Use algebra tosimplify the expression at this step then continue to work onfinding the limit to infinity.
I this works because for large values of x the highest powerterm of the polynomial is so much larger that all of thesmaller degree terms that the smaller degree terms have noeffect in the limit to infinity.
![Page 38: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/38.jpg)
Special Limitsx→ ±∞ for Polynomials
I when taking limits of polynomials to ±∞ drop the lowerdegree terms and only keep the higest degree term of of thepolymomial.
I this is an intermediate step in taking the limit. Use algebra tosimplify the expression at this step then continue to work onfinding the limit to infinity.
I this works because for large values of x the highest powerterm of the polynomial is so much larger that all of thesmaller degree terms that the smaller degree terms have noeffect in the limit to infinity.
![Page 39: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/39.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3
= limx→+∞2x3
4x3 = limx→+∞24 =
limx→+∞12 = 1
2
![Page 40: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/40.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3
= limx→+∞2x3
4x3 = limx→+∞24 =
limx→+∞12 = 1
2
![Page 41: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/41.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3
= limx→+∞24 =
limx→+∞12 = 1
2
![Page 42: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/42.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3 = limx→+∞24
=
limx→+∞12 = 1
2
![Page 43: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/43.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3 = limx→+∞24 =
limx→+∞12
= 12
![Page 44: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/44.jpg)
Special Limitsexamples of x→ ±∞ for Polynomials
I limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3 = limx→+∞24 =
limx→+∞12 = 1
2
![Page 45: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/45.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x3+99x+1000
1+2x2+4x3
= limx→+∞2x3
4x3
= limx→+∞24
= limx→+∞12
= 12
![Page 46: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/46.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3
= limx→+∞24
= limx→+∞12
= 12
![Page 47: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/47.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3
= limx→+∞24
= limx→+∞12
= 12
![Page 48: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/48.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3
= limx→+∞24
= limx→+∞12
= 12
![Page 49: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/49.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x3+99x+1000
1+2x2+4x3 = limx→+∞2x3
4x3
= limx→+∞24
= limx→+∞12
= 12
![Page 50: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/50.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3
= limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 51: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/51.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 52: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/52.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 53: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/53.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 54: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/54.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 55: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/55.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 56: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/56.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→−∞2x4+99x+1000
1+2x2+4x3 = limx→−∞2x4
4x3
= limx→−∞24x
= limx→−∞12x
= 12 limx→−∞ x
= 12 · −∞
= −∞
![Page 57: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/57.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6
= limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 58: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/58.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 59: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/59.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 60: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/60.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 61: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/61.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 62: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/62.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 63: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/63.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 64: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/64.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 65: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/65.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 66: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/66.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 67: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/67.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 68: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/68.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0
![Page 69: Special Limits - homepages.math.uic.edurmlowman/math165/LectureNotes/L4-W2L1speciallimits.pdfSpecial Limits e the natural base I the number e is the natural base in calculus. Many](https://reader033.fdocuments.net/reader033/viewer/2022041616/5e3bee03b23ec26bfa2249fc/html5/thumbnails/69.jpg)
Special Limitsexample of x→ ±∞ for Polynomials
limx→+∞2x4+99x+1000
1+2x2+4x6 = limx→+∞2x4
4x6
= limx→+∞2
4x2
= limx→+∞1
2x2
= 12 limx→+∞
1x2
= 12 ·
1+∞
= 12 · 0
+ = 0+ = 0