Lesson 6: Continuity II, Infinite Limits
-
Upload
matthew-leingang -
Category
Education
-
view
2.776 -
download
0
Transcript of Lesson 6: Continuity II, Infinite Limits
![Page 1: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/1.jpg)
Lesson 6Continuity, Infinite Limits
Math 1a
October 5, 2007
Announcements
I No class Monday 10/8, yes class Friday 10/12
I MQC closed Sunday, open Monday
![Page 2: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/2.jpg)
Illustrating the IVTSuppose that f is continuous on the closed interval [a, b] and let Nbe any number between f (a) and f (b), where f (a) 6= f (b). Thenthere exists a number c in (a, b) such that f (c) = N.
x
f (x)
a b
f (a)
f (b)
N
c
c1 c2 c3
![Page 3: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/3.jpg)
Using the IVT
Example
Let f (x) = x3 − x − 1. Show that there is a zero for f . Estimate itwithin 1/16.
![Page 4: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/4.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page2of11
![Page 5: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/5.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page3of11
![Page 6: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/6.jpg)
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
![Page 7: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/7.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page4of11
![Page 8: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/8.jpg)
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
![Page 9: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/9.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page5of11
![Page 10: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/10.jpg)
Back to the Questions
True or FalseAt one point in your life you were exactly three feet tall.
True or FalseAt one point in your life your height in inches equaled your weightin pounds.
True or FalseRight now there are two points on opposite sides of the Earth withexactly the same temperature.
![Page 11: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/11.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page7of11
![Page 12: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/12.jpg)
Infinite Limits
DefinitionThe notation
limx→a
f (x) =∞
means that the values of f (x) can be made arbitrarily large (aslarge as we please) by taking x sufficiently close to a but not equalto a.
DefinitionThe notation
limx→a
f (x) = −∞
means that the values of f (x) can be made arbitrarily largenegative (as large as we please) by taking x sufficiently close to abut not equal to a.
Of course we have definitions for left- and right-hand infinite limits.
![Page 13: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/13.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page8of11
![Page 14: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/14.jpg)
Vertical Asymptotes
DefinitionThe line x = a is called a vertical asymptote of the curvey = f (x) if at least one of the following is true:
I limx→a f (x) =∞I limx→a+ f (x) =∞I limx→a− f (x) =∞
I limx→a f (x) = −∞I limx→a+ f (x) = −∞I limx→a− f (x) = −∞
![Page 15: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/15.jpg)
Infinite Limits we Know
limx→0+
1
x=∞
limx→0−
1
x= −∞
limx→0
1
x2=∞
![Page 16: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/16.jpg)
Finding limits at trouble spots
Example
Let
f (t) =t2 + 2
t2 − 3t + 2
Find limt→a− f (t) and limt→a+ f (t) for each a at which f is notcontinuous.
SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.
![Page 17: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/17.jpg)
Finding limits at trouble spots
Example
Let
f (t) =t2 + 2
t2 − 3t + 2
Find limt→a− f (t) and limt→a+ f (t) for each a at which f is notcontinuous.
SolutionThe denominator factors as (t − 1)(t − 2). We can record thesigns of the factors on the number line.
![Page 18: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/18.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
![Page 19: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/19.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
![Page 20: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/20.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
![Page 21: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/21.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
![Page 22: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/22.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+
±∞ − ∓∞ +
![Page 23: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/23.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞
− ∓∞ +
![Page 24: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/24.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ −
∓∞ +
![Page 25: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/25.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞
+
![Page 26: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/26.jpg)
(t − 1)−
1
0 +
(t − 2)−
2
0 +
(t2 + 2)+
f (t)1 2
+ ±∞ − ∓∞ +
![Page 27: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/27.jpg)
Math 1a - October 05, 2007.GWBFriday, Oct 5, 2007
Page11of11
![Page 28: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/28.jpg)
Limit Laws with infinite limits
I The sum of positive infinite limits is ∞. That is
∞+∞ =∞
I The sum of negative infinite limits is −∞.
−∞−∞ = −∞
I The sum of a finite limit and an infinite limit is infinite.
a +∞ =∞a−∞ = −∞
![Page 29: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/29.jpg)
Rules of Thumb with infinite limits
I The sum of positive infinite limits is ∞. That is
∞+∞ =∞
I The sum of negative infinite limits is −∞.
−∞−∞ = −∞
I The sum of a finite limit and an infinite limit is infinite.
a +∞ =∞a−∞ = −∞
![Page 30: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/30.jpg)
Rules of Thumb with infinite limitsI The product of a finite limit and an infinite limit is infinite if
the finite limit is not 0.
a · ∞ =
{∞ if a > 0
−∞ if a < 0.
a · (−∞) =
{−∞ if a > 0
∞ if a < 0.
I The product of two infinite limits is infinite.
∞ ·∞ =∞∞ · (−∞) = −∞
(−∞) · (−∞) =∞
I The quotient of a finite limit by an infinite limit is zero:
a
∞= 0.
![Page 31: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/31.jpg)
I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.
I Limits of the form 10 are also indeterminate.
![Page 32: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/32.jpg)
I Limits of the form 0 · ∞ and ∞−∞ are indeterminate. Thereis no rule for evaluating such a form; the limit must beexamined more closely.
I Limits of the form 10 are also indeterminate.
![Page 33: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/33.jpg)
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.
![Page 34: Lesson 6: Continuity II, Infinite Limits](https://reader033.fdocuments.net/reader033/viewer/2022052412/558e30341a28ab4d618b46b3/html5/thumbnails/34.jpg)
Example
Compute limx→∞
(√4x2 + 17− 2x
).
SolutionThis limit is of the form ∞−∞, which we cannot use. So werationalize the numerator (the denominator is 1) to get anexpression that we can use the limit laws on.