Evaluating Limits Analytically

Lesson 1.3

2

What Is the Squeeze Theorem?

Today we look at various properties of limits, including the Squeeze Theorem

Today we look at various properties of limits, including the Squeeze Theorem

3

How do we evaluate limits?

• Numerically– Construct a table of values.

• Graphically– Draw a graph by hand or use TI’s.

• Analytically– Use algebra or calculus.

4

Properties of Limits The Fundamentals

bbcx

lim cxcx

lim nn

cxcx

lim

Basic Limits:

Let b and c be real numbers and

let n be a positive integer:

5

Examples:

3lim2x

x

x 4lim

2

3lim xx

6

Properties of Limits Algebraic Properties

Algebraic Properties of Limits:

Let b and c be real numbers, let n be a positive integer, and let f and g be functions

with the following properties:

Too many to fit on this page….

7

Properties of Limits Algebraic Properties

Lxfcx

)(lim Kxgcx

)(limLet: and

bLxbfcx

)(lim

KLxgxfcx

)()(lim

LKxgxfcx

)()(lim

Scalar Multiple:

Sum or Difference:

Product:

8

Properties of Limits Algebraic Properties

Lxfcx

)(lim Kxgcx

)(limLet: and

Quotient:

Power:

0;)(

)(lim

KK

L

xg

xfcx

nn

cxLxf

)(lim

9

Evaluate by using the properties of limits. Show each step and

which property was used.

34lim 2

2x

x

10

Examples of Direct Substitution - EASY

3

5

9

2

2

2

2

lim

lim3

lim 4 3

1lim

1

x

x

x

x

x

x

x x

x

33

2

2

1 1(5)

5 125

3

4(2) 3 19

2 2 15

2 1

11

Examples

2

7

lim cos

limsec6

x

x

x

x

2 2lim cos ( 1) 1x

3

32

3

2

23

1

67

cos

1

6

7seclim

7

x

12

Properties of Limits nth roots

Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid

for all c > 0 if n is even…

nn

cxcx

lim

13

Properties of Limits Composite Functions

If f and g are functions such that…

Lxgcx

)(lim )()(lim LfxfLx

and

then…

)()(lim)(lim Lfxgfxgfcxcx

14

Example:

3 2

3102lim x

x

By now you should have already arrived at the conclusion that many algebraic functions can be evaluated by direct substitution.

The six basic trig functions also exhibit this desirable characteristic…

15

Properties of Limits Six Basic Trig Function

Let c be a real number in the domain of the

given trig function.

cxcx

sinsinlim

cxcx

coscoslim

cxcx

tantanlim

cxcx

secseclim

cxcx

csccsclim

cxcx

cotcotlim

16

A Strategy For Finding Limits

• Learn to recognize which limits can be evaluated by direct substitution.

• If the limit of f(x) as x approaches c cannot be evaluated by direct substitution, try to find a function g that agrees with f for all x other than x = c.

• Use a graph or table to find, check or reinforce your answer.

17

The Squeeze Theorem

and

)()( xgxf

ba,

bca

)(lim)(lim xgxfcxcx

FACT:If

for all x on

then,

18

Example:

xx

x

1coslim 2

0

GI-NORMOUS PROBLEMS!!!

Use Squeeze Theorem!

19

1cos1 x

11

cos1 x

222 1cos x

xxx

0lim 2

0

x

x0lim 2

0

x

x

01

coslim 2

0

xx

x

20

21

Example:Use the squeeze theorem to find:

22 4)(4 xxfx

)(lim0

xfx

22

Properties of Limits Two Special Trig Function

1sin

lim0

x

xx

0cos1

lim0

x

xx

23

General Strategies

24

Some Examples

• Consider

– Why is this difficult?

• Strategy: simplify the algebraic fraction

2

2

6lim

2x

x x

x

2

2 2

2 36lim lim

2 2x x

x xx x

x x

25

Reinforce Your Conclusion

• Graph the Function– Trace value close to

specified point

• Use a table to evaluateclose to the point inquestion

26

Find each limit, if it exists.3

1

11. lim

1x

x

x

27

Find each limit, if it exists.3

1

11. lim

1x

x

x

2

1

1 1lim

1x

x x x

x

2

1lim 1x

x x

3

Direct Substitution doesn’t work!

Factor, cancel, and try again!

D.S.

Don’t forget, limits can never be undefined!

28

Find each limit, if it exists.

0

1 12. lim

x

x

x

29

Find each limit, if it exists.

0

1 12. lim

x

x

x

1 1x

1 1x

Direct Substitution doesn’t work.

Rationalize the numerator.

0

1 1lim

1 1x

x

x x

0lim

1 1x

x

x x

0

1lim

1 1x x

1

2

D.S.

30

Special Trig Limits

0

tan43. lim

x

x

x

0

sinlimx

Ax

Ax

0

1 coslimx

Ax

Ax

0

cos 1limx

Ax

Ax

31

Special Trig Limits

0

tan43. lim

x

x

x

0

sinlimx

Ax

Ax

0

1 coslimx

Ax

Ax

1 0

0

cos 1limx

Ax

Ax

0

0

sin4 1lim

cos4x

x

x x

0

sin4 1lim

cos4x

x

x x

0

sin4 4lim

4 cos4x

x

x x

4

4

0 0

sin4 4lim lim

4 cos4x x

x

x x

1 4 4Trig limit

D.S.

32

Evaluate in any way you chose.

3

6lim

2

3 x

xxx

33

Evaluate in any way you chose.

2

3 3

6 ( 3)( 2)lim lim

3 3x x

x x x x

x x

34

Evaluate in any way you chose.

2

3 3

3

6 ( 3)( 2)lim lim

3 ( 3)

lim( 2)

x x

x

x x x x

x x

x

35

Evaluate in any way you chose.

2

3 3

3

6 ( 3)( 2)lim lim

3 ( 3)

lim( 2) 5

x x

x

x x x x

x x

x

36

Evaluate by using a graph.

Is there a better way?

x

xx

11lim

0

37

0

1 1 1 1lim

1 1x

x x

x x

38

0

0

1 1 1 1lim

1 1

1 1lim

( 1 1)

x

x

x x

x x

x

x x

39

0

0 0

1 1 1 1lim

1 1

1 1lim lim

( 1 1) ( 1 1)

x

x x

x x

x x

x x

x x x x

40

0

0 0 0

1 1 1 1lim

1 1

1 1 1lim lim lim

( 1 1) ( 1 1) ( 1 1)

x

x x x

x x

x x

x x

x x x x x

41

0

0 0 0

1 1 1 1lim

1 1

1 1 1lim lim lim

( 1 1) ( 1 1) ( 1 1)

1

2

x

x x x

x x

x x

x x

x x x x x

42

Evaluate:

x

xx

3

1

3

1

lim0

43

Evaluate:

0 0

3 31 13(3 )3 3lim lim

x x

xxx

x x

44

Evaluate:

0

3 31 13(3 )3 3lim

3 (3 )x

xxxx

x x x x

45

Evaluate:

0 0 0

0

3 31 13(3 )3 3lim lim lim

3 (3 )

1 1lim

3(3 ) 9

x x x

x

xxxx

x x x x

x

46

Evaluate:4

2

16lim

2x

x

x

47

Evaluate:4

2 2

16 ( 2)( 2)lim lim

2 2x x

x x x

x x

48

Evaluate:4

2 2

2

16 ( 2)( 2)lim lim

2 ( 2)

lim( 2)

x x

x

x x x

x x

x

49

Evaluate:4

2 2

2

16 ( 2)( 2)lim lim

2 ( 2)

lim( 2) 4

x x

x

x x x

x x

x

50

Evaluate:

h

hh

1832lim

2

0

51

Evaluate:

2 2

0 0

2 3 18 2(9 6 ) 18lim limh h

h h h

h h

52

Evaluate:

2 2

0 0

2

0

2 3 18 2(9 6 ) 18lim lim

18 12 2 18lim

h h

h

h h h

h h

h h

h

53

Evaluate:

2 2

0 0

2 2

0 0 0

2 3 18 2(9 6 ) 18lim lim

18 12 2 18 12 2 2 (6 )lim lim lim

h h

h h h

h h h

h h

h h h h h h

h h h

54

Evaluate:

2 2

0 0

2 2

0 0 0

0

2 3 18 2(9 6 ) 18lim lim

18 12 2 18 12 2 2 (6 )lim lim lim

lim 2(6 )

h h

h h h

h

h h h

h h

h h h h h h

h h hh

55

Evaluate:

2 2

0 0

2 2

0 0 0

0

2 3 18 2(9 6 ) 18lim lim

18 12 2 18 12 2 2 (6 )lim lim lim

lim 2(6 ) 12

h h

h h h

h

h h h

h h

h h h h h h

h h hh

56

Evaluate:

57

• Note possibilities for piecewise defined functions. Does the limit exist?

2

2

3 2 2( )

5 2

lim ( ) ?x

x if xf x

x if x

f x

58

Three Special Limits

• Try it out!

0

sin 4lim ?

9x

x

x 20

1 coslimx

x

x

1

0 0 0

sin 1 coslim 1 lim 0 lim 1 xx x x

x xx e

x x

59

x

xx 7sin

9sinlim

0

xxxx

x

xx

7799

7sin

9sinlim

0

xx

x

xx

x

x

77sin

7

99sin

9lim

0

xxxx

x

x

77sin

lim

99sin

lim

7

9

0

0

7

9

1

1

7

9

60

Squeeze Rule

• Given g(x) ≤ f(x) ≤ h(x) on an open interval containing cAnd …

– Then

lim ( ) lim ( )

lim ( )

x c x c

x c

g x h x L

f x L

61

Common Types of Behavior Associated with the Nonexistence of

a Limit1. f(x) approaches a different number from

the right side of c than it approaches from the left side.

2. f(x) increases or decreases without bound as x approaches c.

3. f(x) oscillates between 2 fixed values as x approaches c.

62

Gap in graph Asymptote

Oscillates

c c

c

existnotdoescx

lim

existnotdoescx

lim