1.5: Limits Involving Infinity

Learning Goals:

©2009 Mark Pickering

•Calculate limits as

•Identify vertical and horizontal asymptotes

x

Example

Find the limit if it exists:2

3

6lim

3x

x x

x

In the previous lesson, we had this problem:

Example

Find the limit if it exists:2 6

lim3x

x x

x

How does this problem differ from the previous problem?

Important Idea, ,

The above symbols describe the increasing or decreasing of a value without bound. Infinity is not a number.

ExampleUse the table feature of your calculator to estimate:

if it exists.

4 5lim

2 8x

x

x

Definition

if they exist, means y=b is a horizontal asymptote

b

or

lim ( ) lim ( )x x

f x b f x b

2

1( ) 2f x

x

Try ThisUse the graph and table features of your calculator to find any horizontal asymptotes for:

2y

Try ThisUse the graph and table features of your calculator to find any horizontal asymptotes for:

2( )

1

xf x

x

1y & 1y

Important IdeaFunctions involving radicals may have 2 horizontal asymptotes

Horizontal asymptotes are always written as

y

2

2

2 1lim

3 1

x

xx

ExampleFind the limit, if it exists:

Indeter-minate form

Divide top & bottom by highest power of x in denominator.

3

2

3 5lim

3 1

x

xx

Try ThisFind the limit, if it exists:

DNE or

2

3

3 1lim

3 5

x

xx

Try ThisFind the limit, if it exists:

0

2

2

2 1lim

3 5

x

xx

Try ThisFind the limit, if it exists:

2

3

2

2

2 1 2lim

33 5

x

xx

Analysis

2

3

3 10lim

3 5

x

xx

3

2

3 5lim

3 1

x

xx

In the last

3 examples, do you see a pattern?The highest power term is most influential.

Important Idea

•If the degree of the top is greater than the degree of the bottom,

For any rational function( )

( )( )

g xf x

p x

lim ( )x

f x

Important Idea

•If the degree of the bottom is greater than the degree of the top,

For any rational function( )

( )( )

g xf x

p x

lim ( ) 0x

f x

Important Idea

•If the degree of the top is the same as the degree of the top, the limit as

For any rational function( )

( )( )

g xf x

p x

x is the ratio of the leading coefficients.

Try ThisFind the limit if it exists: 3

33

lim2 5 2x

x x

x x

3

2

ExampleFunctions may approach different asymptotes as x and

asx 2 2

lim3 6

x

xx

Consider each limit separately…

2 2lim

3 6

x

xx

Example

2x xAs eventually x >0. Divide radical by and divide non-radical by x.

Find the limit:

x

2 2lim

3 6

x

xx

Example

2x xAs eventually x <0. Divide radical by and divide non-radical by x.

Find the limit:

x

Try This

2

3 1lim

x

x xx

Using algebraic techniques, find the limit if it exists. Confirm your answer with your calculator.

Solution

2

3 13lim

x

x xx

Analysis

The sine function oscillates between +1 and -1

1

-1

Analysis

1

-1

sinlim xx

The limit does not exist due to oscillation.

Analysissin

limx

xx

Consider

1 sin 1x

1 sin 1 x

x x x

Analysis1

0limxx

10lim

xx

and

Therefore, by the Sandwich Theorem,

sin0lim

x

xx

A function has an infinite limit at a if f(a) as xa. f(x) is unbounded at x=a.

f(x)

lim ( )x a

f x

lim ( )x a

f x

Definition

A function has an infinite limit at a if f(a) as xa. f(x) is unbounded at x=a.

f(x)

Definition

The line x=a is a vertical asymptote

Important Idea

A vertical asymptote is written “x=a”

A horizontal asymptote is written “y=a”

where a is any real number.

Try ThisWhat is the limit as x1 from the left and from the right?

x=1What is the vertical asymptote?

Solution

x=1

1lim ( )x

f x

1lim ( )x

f x

Vertical asymptote: x=1

Try ThisWhat is the limit as x1 from the left and from the right?

x=1What is the vert. asymptote?

Solution

1lim ( )x

f x

x=11

lim ( )x

f x

Vertical asymptote: x=1

DefinitionThe value(s) that make the denominator of a rational function zero is a vertical asymptote.

ExampleDetermine all vertical asymptotes of

2

2

2 8( )

4

x xf x

x

Steps:1. Factor & cancel if possible2. Set denominator to 0 & solve

ExampleFind the limit if it exists:

2

1

3lim

1x

x x

x

1. When you substitute x=1, do you get a number/0 or 0/0?

The questions…

2. What is happening at x= a value larger than 1?

Try This

Find the limit if it exists: 2

1

3lim

1x

x x

x

+

Find all vertical asymptotes for for

( ) tanf 0 2

Hint: since ,

Example

sin( )

cosf

where does ?cos 0

Lesson Close

Limits are the foundation of both differential and integral Calculus. We will develop these ideas in chapter 3.

Practice

76/1-7,13,15,27-33,59 (all odd)