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