1 Mrs. Kessler Calculus Finding Limits Analytically 1.3 part 2 Calculus.

1Mrs. Kessler

CalculusFinding Limits

Analytically1.3 part 2

Calculus

2

0lim 4x

x

2

lim

lim

lim

x c

x c

n n

x c

b b

x c

x c

®

®

®

=

=

=

Basic Properties

3

Limit Properties

4

Thank you for not dividing by zero.

• What happens when you "sub in" the value of c in the and the denominator equals zero???

lim ( )x c

f x

2

25

3 10lim

25x

x x

x

For example, this limit.

5

New Techniques to find LimitsNew Techniques to find Limits

2. Rationalizing the numerator

1. Dividing out

3. Special cases

6

Dividing Out Technique: Factor, then reduce.

5

( 5)( 2)lim

( 5)( 5)x

x x

x x

2

25

3 10lim

25x

x x

x

= 7

10

Since we are taking the limit as x approaches 5, and not at x = 5, we do not have to worry about dividing by zero.

Example 1:

7

Dividing Out Technique: Factor, then reduce

0

( 1)( 1)lim

( 1)

x x

xx

e e

e

2

0

1lim

1

x

xx

e

e

2

Since we are again taking the limit as x approaches 0, and not at x = 0, we do not have to worry about dividing by zero.

Direct substitution yields the indeterminate form 0/0.

Factor

Example 2:

8

Rationalizing Technique

0

3 3 3 3lim

3 3x

x x

x x

0

3 3lim

3 3x

x

x x

0

3 3limx

x

x

0

lim3 3x

x

x x

1 3

62 3

We rationalize the numerator instead of the denominator. We are still multiplying by one, thereby not changing the value, just the look.

Example 3:

9

7

2

128lim

2x

x

x

What happens when you substitute x = 2?

Use synthetic to simplify and divide.

6 5 4 3 2

2lim 2 4 8 16 32 64x

x x x x x x

448

Example 8:

10

Transcendental LimitsTranscendental Limits

limsin sin limcsc csc

limcos cos limsec sec

lim tan tan limcot cot

lim limln ln

x c x c

x c x c

x c x c

x x

x c x c

x c x c

x c x c

x c x c

a c x c

11

Special Cases

Theorem 1. 8 The Squeeze TheoremTheorem 1. 8 The Squeeze Theorem

If h(x) < f(x) < g(x) for all x in an open interval containing c, except possible at c itself, and if

lim ( ) lim ( )

then lim ( ) exists and is equal to L.x c x c

x c

h x L g x

f x

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…

ExampleFind the limit if it exists:

0

sinlim

Factor and cancel doesn’t work…

ExampleFind the limit if it exists:

0

sinlim

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

19

0

sinlim 1x

x

x

0

1 coslim 0x

x

x

1/

0lim(1 ) x

xx e

The proof is in the book, and uses the squeeze theorem.

You must learn these!

Special limits whose proofs use the squeeze theorem

20

Example 4: 2

0

tanlimx

x

x

Rewrite2

20

sinlim

cosx

x

x x

20

sin sinlim

cosx

x x

x x

20 0

sin sinlim lim

cosx x

x x

x x

0

20

sinlim 1

sin 0and lim

cos 1

x

x

Since

x

x

x

x

= (1)(0) = 0

21

0

(3 3cos )limx

x

x

Direct substitution gives 0/0 which is indeterminate. Rewrite.

0

(1 cos )3lim

x

x

x

0

(1 cos )3lim 3(0) 0

x

x

x

0

(1 cos )lim 0x

xSince

x

Example 5:

22

0

sin5limx

x

x

Multiply the numerator and the denominator by 5.

0

sin5lim

5

5x

x

x 0

s5

ili

n5

5m

x

x

x

= 5(1) = 5

Special case

Example 6:

23

0

3 tan 3lim

3x

cos x x

x

Rewrite

0

cos3 sin 3lim

cos3 3x

x x

x x

0

sin 3lim

3x

x

x1

Example 7:

24

Sometimes you have to be creative when determining which method to use and rely

upon all previous mathematics.