Chapter 2 2012 Pearson Education, Inc. 2.2 Limits Involving Infinity Section 2.2 Limits and Continuity.

2012 Pearson Education, Inc.

Chapter 2

2.2Limits Involving Infinity

Section 2.2

Limits and Continuity

Slide 2.2- 2 2012 Pearson Education, Inc.

Quick Review

1 1In Exercises 1– 4, find and graph , and in the

same viewing window.

1. 2 3 2. x

f f f y x

f x x f x e

Slide 2.2- 3 2012 Pearson Education, Inc.

Quick Review

1 13. tan 4. cotf x x f x x

Slide 2.2- 4 2012 Pearson Education, Inc.

Quick Review

3 2 3

5 3 3 2

In Exercises 5 and 6, find the quotient and remainder

when is divided by .

5. 2 3 1, 3 4 5

6. 2 1, 1

q x r x

f x g x

f x x x x g x x x

f x x x x g x x x

Slide 2.2- 5 2012 Pearson Education, Inc.

Quick Review

1In Exercises 7 –10, write a formula for a and b .

Simplify where possible.

7. cos 8.

ln 19. 10. sin

x

f x fx

f x x f x e

xf x f x x x

x x

Slide 2.2- 6 2012 Pearson Education, Inc.

Quick Review Solutions

1

1

1

1In Exercises 1– 4, find and graph , and in the

same viewing window.

1.

3ln

2

2 3 2. x

f f f y x

f

xf

x e

f

f

x

x

x x

x

[12,12] by [8,8] [6,6] by [4,4]

Slide 2.2- 7 2012 Pearson Education, Inc.

Quick Review Solutions

, by ,3 3 2 2

0, by 1,

1 1

1 13. tan 4. cot

tan cot

02 2

f x x f x

f x x f

x

x

x

x

x

Slide 2.2- 8 2012 Pearson Education, Inc.

Quick Review Solutions

3 2 3

5 3 3 2

2

2 2

In Exercises 5 and 6, find the quotient and remainder

when is divided by .

5. 2 3 1, 3 4

2 5 7, 3

3 3 3

2 2 1,

5

6. 2

2

1, 1

q x r x

f x g x

f x x x x

q

g x x x

f x x x x g x

x r x

x

x x

q x x x r x x x

x

Slide 2.2- 9 2012 Pearson Education, Inc.

Quick Review Solutions

11 1 1cos , cos ,

ln 1 1,

1In Exercises 7 –10, write a formula for a and b .

Simplify where possible.

7. cos 8.

lln

9.

110. s

n

x x

x

f x x f f x e f ex x x

xf x f x

x x x

f x fx

f x x f x e

xf x

x

f x xx

1 1 1 1sin , sin ni f x x x f x

x x xx

x

Slide 2.2- 10 2012 Pearson Education, Inc.

What you’ll learn about

Finite Limits as x→±∞ Sandwich Theorem Revisited Infinite Limits as x→a End Behavior Models Seeing Limits as x→±∞

…and whyLimits can be used to describe the behavior of functionsfor numbers large in absolute value.

Slide 2.2- 11 2012 Pearson Education, Inc.

Finite limits as x→±∞

The symbol for infinity (∞) does not represent a real number. We use ∞ to describe the behavior of a function when the values in its domain or range outgrow all finite bounds.

For example, when we say “the limit of f as x approaches infinity” we mean the limit of f as x moves increasingly far to the right on the number line.

When we say “the limit of f as x approaches negative infinity (∞)” we mean the limit of f as x moves increasingly far to the left on the number line.

Slide 2.2- 12 2012 Pearson Education, Inc.

Horizontal Asymptote

The line is a of the graph of a function

if either

lim or limx x

y b

y f x

f x b f x b

horizontal asymptote

Slide 2.2- 13 2012 Pearson Education, Inc.

[-6,6] by [-5,5]

Example Horizontal Asymptote

1xf x

x

a lim 1x

f x

b lim 1x

f x

c Identify all horizontal asymptotes. 1y

c Identify all horizontal asymptotes.

Use a graph and tables to find a lim and b lim .x x

f x f x

Slide 2.2- 14 2012 Pearson Education, Inc.

Example Sandwich Theorem Revisited

The sandwich theorem also holds for limits as . x

cosFind lim graphically and using a table of values.

x

x

x

The graph and table suggest that the function oscillates about the -axis.x

cosThus 0 is the horizontal asymptote and lim 0

x

xy

x

Slide 2.2- 15 2012 Pearson Education, Inc.

If , and are real numbers and

lim and lim , then

1. : lim

The limit of the sum of two functions is the sum of their limits.

2. : lim

The limi

x x

x

x

L M k

f x L g x M

Sum Rule f x g x L M

Difference Rule f x g x L M

t of the difference of two functions is the difference

of their limits

Properties of Limits as x→±∞

Slide 2.2- 16 2012 Pearson Education, Inc.

Properties of Limits as x→±∞

3. : lim

The limit of the product of two functions is the product of their limits.

4. : lim

The limit of a constant times a

x

x

Product Rule f x g x L M

Constant Multiple Rule k f x k L

function is the constant times the limit

of the function.

5. : lim , 0

The limit of the quotient of two functions is the quotient

of their limits, provi

x

f x LQuotient Rule M

g x M

ded the limit of the denominator is not zero.

Slide 2.2- 17 2012 Pearson Education, Inc.

Properties of Limits as x→±∞

6. : If and are integers, 0, then

lim

provided that is a real number.

The limit of a rational power of a function is that power of the

limit of the function, provided the latt

rrss

x

r

s

Power Rule r s s

f x L

L

er is a real number.

Slide 2.2- 18 2012 Pearson Education, Inc.

Infinite Limits as x→a

If the values of a function ( ) outgrow all positive bounds as approaches

a finite number , we say that lim . If the values of become large

and negative, exceeding all negative bounds as x a

f x x

a f x f

approaches a finite number ,

we say that lim . x a

x a

f x

Slide 2.2- 19 2012 Pearson Education, Inc.

Vertical Asymptote

The line is a of the graph of a function

if either

lim or lim x a x a

x a

y f x

f x f x

vertical asymptote

Slide 2.2- 20 2012 Pearson Education, Inc.

Example Vertical Asymptote

[6,6] by [6,6]

Find the vertical asymptotes of the graph of ( ) and describe the behavior

of ( ) to the right and left of each vertical asymptote.

f x

f x

2

8

4f x

x

The values of the function approach to the left of 2.x The values of the function approach + to the right of 2.x The values of the function approach + to the left of 2.x The values of the function approach to the right of 2.x

2 22 2

8 8lim and lim

4 4x xx x

2 22 2

8 8lim and lim

4 4x xx x

So, the vertical asymptotes are 2 and 2x x

Slide 2.2- 21 2012 Pearson Education, Inc.

End Behavior Models

The function is

a a for if and only if lim 1.

b a for if and only if lim 1.

x

x

g

f xf

g x

f xf

g x

right end behavior model

left end behavior model

Slide 2.2- 22 2012 Pearson Education, Inc.

Example End Behavior Models

Find an end behavior model for

2

2

3 2 5

4 7

x xf x

x

2

2

Notice that 3 is an end behavior model for the numerator of , and

4 is one for the denominator. This makes

x f

x2

2

3 3= an end behavior model for .

44

xf

x

Slide 2.2- 23 2012 Pearson Education, Inc.

End Behavior Models

1

If one function provides both a left and right end behavior model, it is simply

called an .

In general, is an end behavior model for the polynomial function nn

n nn n

g x a x

f x a x a x

end behavior model

10... , 0

Overall, all polynomials behave like monomials.na a

Slide 2.2- 24 2012 Pearson Education, Inc.

End Behavior Models

3In this example, the end behavior model for , is also a horizontal

4asymptote of the graph of . We can use the end behavior model of a

rational function to identify any horizontal asymptote.

A

f y

f

rational function always has a simple power function as

an end behavior model.

Slide 2.2- 25 2012 Pearson Education, Inc.

Example “Seeing” Limits as x→±∞

We can investigate the graph of as by investigating the y f x x

1graph of as 0.

y f xx

1Use the graph of to find lim and lim

x x

y f f x f xx

1for cos .f x x

x

1 cosThe graph of = is shown.

xy f

x x

0

1lim lim

x x

f x fx

0

1lim lim

x x

f x fx

Slide 2.2- 26 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

2

3

You may use a graphing calculator to solve the following problems.

61. Find lim if it exists

3

A 1

B 1

C 2

D 5

E does not exist

x

x x

x

Slide 2.2- 27 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

2

3

You may use a graphing calculator to solve the following problems.

61. Find lim if it exists

3

A 1

B 1

C 2

E does not exist

D 5

x

x x

x

Slide 2.2- 28 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

2

3 1, 22. Find lim = if it exists5

, 21

5A

313

B 3

C 7

D

E does not exist

x

x xf x

xx

Slide 2.2- 29 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

2

3 1, 22. Find lim = if it exists5

, 21

13B

3C

5A

3

7

D

E does not exist

x

x xf x

xx

Slide 2.2- 30 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

3 2

3

3. Which of the following lines is a horizontal asymptote for

3 7

2 4 53

A2

B 0

2C

37

D53

E2

x x xf x

x x

y x

y

y

y

y

Slide 2.2- 31 2012 Pearson Education, Inc.

Quick Quiz Sections 2.1 and 2.2

3 2

3

3. Which of the following lines is a horizontal asymptote for

3 7

2 4 53

A2

B 0

2C

37

D

3E

5

2

x x xf x

x x

y x

y

y

y

y