3.5 Limits Involving Infinity North Dakota Sunset.

3.5 Limits Involving Infinity

North Dakota Sunset

What you’ll learn aboutFinite Limits as x→±∞Sandwich Theorem RevisitedInfinite Limits as x→aEnd Behavior Models Seeing Limits as x→±∞

…and whyLimits can be used to describe the behavior

of functions for numbers large in absolute value.

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.

Horizontal Asymptote

The line is a of the graph of a function if either

lim or limx x

y by f x

f x b f x b

horizontal asymptote

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

Example Horizontal Asymptote

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

c Identify all horizontal asymptotes.

1

x xf x f x

xf xx

a lim 1

b lim 1

c Identify all horizontal asymptotes. 1

x

x

f x

f x

y

1f xx

1lim 0x x

As the denominator gets larger, the value of the fraction gets smaller.

There is a horizontal asymptote if:

limx

f x b

or limx

f x b

2lim

1x

x

x 2limx

x

x

This number becomes insignificant as .x

limx

xx

1

There is a horizontal asymptote at 1.

5 sinlimx

x xx

Find:

5 sinlimx

x xx x

sinlim 5 limx x

xx

5 0

5

Infinite Limits:

1f xx

0

1limx x

As the denominator approaches zero, the value of the fraction gets very large.

If the denominator is positive then the fraction is positive.

0

1limx x

If the denominator is negative then the fraction is negative.

vertical asymptote at x=0.

20

1limx x

20

1limx x

The denominator is positive in both cases, so the limit is the same.

20

1 limx x

Often you can just “think through” limits.

1lim sinx x

00

lim sinx

x

0

Quick Quiz

2

3

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

61. Find lim if it exists3

A 1B 1C 2

E does not exiD 5

st

x

x xx

Slide 2- 12

Quick Quiz

2

3 1, 22. Find lim = if it exists5 , 2

15A313B3

C 7DE does not exist

x

x xf x

xx

Slide 2- 13

Quick Quiz

2

3 1, 22. Find lim = if it ex

5A

ists5 , 21

13B

3C 7DE does not e

3

xist

x

x xf x

xx

Slide 2- 14

Quick Quiz

3 2

3

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

3 7 2 4 5

3A2

B 0

2C3

7D5

3E2

x x xf xx x

y x

y

y

y

y

Slide 2- 15

Quick Quiz

3 2

3

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

3 7 2 4 5

3A2

B 0

2C3

7D

3E

5

2

x x xf xx x

y x

y

y

y

y

Slide 2- 16

PracticeTEXT•p. 88, #9 – 52;•p.89, # 59, 62, 63

•p. 205, #3 – 8, 15 – 20, 21 – 33 odds, 86 – 88.

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