2-5: Techniques for Evaluating Limits ©2002 Roy L. Gover (www.mrgover.com) Objectiv es: •Find limits using direct substitution •Find limits when substitution doesn’t work
Mar 27, 2015
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
Basic Limits (on handout)
limx c
b b
limx c
x c
lim n n
x cx c
0integeran is , :Assume nc
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
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)
Important Idea•The limit, if it exists, of f(x) as xc is f(c) if f(x) is continuous at c.←use substitution
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
Example
Find the limit, if it exists: 3
3lim 2 1x
x
= 2(27) + 1 = 55
Try This
Find the limit, if it exists:
2
2lim 1x
x x
3
Try This
Find the limit, if it exists:
3
4lim 2x
x
-2
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
Important Idea
If substitution results in an a/0 fraction
where a0, the limit doesn’t exist.
There is
no
hope…Horrible
Occurrence!!!
DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.
There is
HOPE!!
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
Important Idea3 21 ( 1)( 1)
1 1
x x x x
x x
2 1x x and
are the same except at x=-1
Important Idea
The functions have the same limit as x-1
Procedure1.Try substitution2. Factor and cancel if
substitution doesn’t work
3.Try substitution again
The factor & cancellation technique
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?
Try ThisFind the limit if it exists:
22
2lim
4x
x
x
1
4
Try This
Find the limit if it exists:2
3
6lim
3x
x x
x
The limit doesn’t existConfirm by graphing
Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as the factor and cancel technique.
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!!
Example
Find the limit, if it exists: 3 5
lim6 8x
x
x
Example
Find the limit, if it exists: 2
3
4lim
2 5x
x x
x
Example
Find the limit, if it exists: 7
3
3 100lim
2 5x
x x
x
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
Try ThisFind the limit if it exists:
0
2 2limx
x
x
1
2 2