Warm-Up

1-3: Evaluating Limits Analytically

©2002 Roy L. Gover (www.mrgover.com)

Objectives:•Find limits when substitution doesn’t work•Learn about the Squeeze Theorem

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

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 againThe 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 Idea

If substitution results in an a/0 fraction

where a0, the limit doesn’t exist.

DefinitionWhen substitution results in a 0/0 fraction, the result is called an indeterminate form.

Important IdeaThe limit of an indeterminate form exists, but to find it you must use a technique, such as the factor and cancel technique.

Try ThisFind the limit if it exists:

2

1

2 3lim

1x

x x

x

-5

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

BC warm-UpFind the limit if it exists:

0

1 1limx

x

x

Try ThisFind the limit if it exists:

1

2 2

0

2 2limx

x

x

The Squeeze Theorem

Let f(x) be between g(x) & h(x) in an interval containing c. Iflim ( ) lim ( )

x c x cg x h x L

lim ( )x cf x L

then:

f(x) is “squeezed” to L

ExampleFind the limit if it exists:

0

sinlim

Where is in radians and in the interval,2 2

ExampleFind the limit if it exists:

0

sinlim

Substitution gives the indeterminate form…Factor and cancel or rationalization doesn’t work…

Maybe…the squeeze theorem…

Example

g()=1

h()=cos

sin( )f

Example

0lim1 1

0

lim cos 1

&

therefore…

0

sinlim 1

Two Special Trig Limits

0

sinlim 1

0

1 coslim 0

Memoriz

e

Example

Find the limit if it exists:

0

tanlimx

x

x

0 0

sin 1lim lim 1 1 1

cosx x

x

x x

Example

Find the limit if it exists:

0

sin(5 )limx

x

x

0 0

sin(5 ) sin(5 )lim 5 5 lim 5 1 5

5 5x x

x x

x x

Try This

Find the limit if it exists:

0

3 3coslimx

x

x

0

Lesson Close

Write, in outline form, the procedures for finding limits when substitution doesn’t work.

Assignment

68/45 – 61 odd