![Page 1: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/1.jpg)
2-5: Techniques for Evaluating Limits
©2002 Roy L. Gover (www.mrgover.com)
Objectives:•Find limits using direct substitution
•Find limits when substitution doesn’t work
![Page 2: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/2.jpg)
Basic Limits (on handout)
limx c
b b
limx c
x c
lim n n
x cx c
0integeran is , :Assume nc
![Page 3: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/3.jpg)
Properties of Limits (on
handout)
KxgLxfnccxcx
)(lim ,)(lim ,0integeran is ,b, :Assume
lim ( ) ( )x c
f x g x L K
lim ( ) ( )x c
f x g x LK
lim ( )x c
b f x bL
( )lim
( )x c
f x L
g x K
lim ( )n n
x cf x L
![Page 4: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/4.jpg)
Properties of Limits (on
handout)
0integeran is ,b, :Assume nc
lim , if 0n n
x cx c c
lim , for all n n
x cx c c
(n is even)
(n is odd)
![Page 5: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/5.jpg)
Important Idea•The limit, if it exists, of f(x) as xc is f(c) if f(x) is continuous at c.←use substitution
![Page 6: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/6.jpg)
Example
2lim ( ) (2) 4x
f x f
f(x) continuous at x=2 f(2) exists and
f(2)=4
The limit is found by substitution
![Page 7: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/7.jpg)
Example
Find the limit, if it exists: 3
3lim 2 1x
x
= 2(27) + 1 = 55
![Page 8: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/8.jpg)
Try This
Find the limit, if it exists:
2
2lim 1x
x x
3
![Page 9: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/9.jpg)
Try This
Find the limit, if it exists:
3
4lim 2x
x
-2
![Page 10: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/10.jpg)
Important Idea•The limit, if it exists, of f(x) as xc is not f(c) if f(x) is discontinuous at c.
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑Cannot use substitution!!Must be other methods
![Page 11: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/11.jpg)
Important Idea
If substitution results in an a/0 fraction
where a0, the limit doesn’t exist.
There is
no
hope…Horrible
Occurrence!!!
![Page 12: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/12.jpg)
DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.
There is
HOPE!!
![Page 13: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/13.jpg)
Example
Find the limit if it exists:3
1
1lim
1x
x
x
Try substitution
Substitution doesn’t work…does this mean the limit doesn’t exist?
Try the factor and cancellation technique
Go to Derive
![Page 14: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/14.jpg)
Important Idea3 21 ( 1)( 1)
1 1
x x x x
x x
2 1x x and
are the same except at x=-1
![Page 15: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/15.jpg)
Important Idea
The functions have the same limit as x-1
![Page 16: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/16.jpg)
Procedure1.Try substitution2. Factor and cancel if
substitution doesn’t work
3.Try substitution again
The factor & cancellation technique
![Page 17: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/17.jpg)
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
5Isn
’t th
at
easy?
Did you think ca
lculus
was going to
be
difficu
lt?
![Page 18: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/18.jpg)
Try ThisFind the limit if it exists:
22
2lim
4x
x
x
1
4
![Page 19: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/19.jpg)
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
The limit doesn’t existConfirm by graphing
![Page 20: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/20.jpg)
Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as the factor and cancel technique.
![Page 21: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/21.jpg)
11 1 0
11 1 0
...lim lim
...
n n nn n n
m m mx xm m m
c x c x c x c c x
d x d x d x d d x
Limits as x→∞, x →-∞
11 1 0
11 1 0
...lim lim
...
n n nn n n
m m mx xm m m
c x c x c x c c x
d x d x d x d d x
Follow the rules for Horizontal Asymptotes!!
![Page 22: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/22.jpg)
Example
Find the limit, if it exists: 3 5
lim6 8x
x
x
![Page 23: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/23.jpg)
Example
Find the limit, if it exists: 2
3
4lim
2 5x
x x
x
![Page 24: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/24.jpg)
Example
Find the limit, if it exists: 7
3
3 100lim
2 5x
x x
x
![Page 25: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/25.jpg)
ExampleFind the limit if it exists:
0
1 1limx
x
x
Try substitutionWith substitution, you get an indeterminate form
Try factor & cancelFactor & cancel doesn’t workHorrib
le
Occurrence!!!
The rationalization technique to the rescue…
Rationalizing the numerator allows you to factor & cancel and then substitute
![Page 26: 2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover () Objectives: Find limits using direct substitution Find limits when substitution](https://reader031.vdocuments.mx/reader031/viewer/2022013003/5514d459550346b0338b525a/html5/thumbnails/26.jpg)
Try ThisFind the limit if it exists:
0
2 2limx
x
x
1
2 2