2.2: LIMITS INVOLVING INFINITY

Objectives:• Students will be able to evaluate limits as • Students will be able to find horizontal and vertical

asymptotes• Students will be able to evaluate infinite limits

x

Finite Limits as

Given f(x) = 1/x

xx

1lim

xx

1lim

x

Look at the graph and table of values for the graph of

What is ?

What else does this tell us?

1

3)(

2

2

x

xxf

)(lim xfx

Definition

The line y=b is a horizontal asymptote of the graph of a function y= f(x) if either

OR

(Note…a graph can have at most 2 HA’s)

bxfx

)(lim bxfx

)(lim

The properties of limits as x ±∞ are on p. 67(same as properties of other limits)Evaluate the limit. Identify any horizontal asymptotes.

xx

12lim

xx

12lim

Theorem

r is a positive #, c is any real #

0lim

0lim

rx

rx

x

c

x

c

Evaluate

1.

2.

1

23lim

25lim

2

x

x

x

x

x

Uh oh…we have . This is indeterminate form. What do we do???

To find finite limits in rational functions…..Divide both the numerator and the denominator by the highest power of x in the denominator. Want to get numerator and denominator in the form then evaluate limitrx

c

Evaluate the limit. Identify the HA.

13

52lim

13

52lim

13

52lim

2

3

2

2

2

x

x

x

x

x

x

x

x

x

Extra examples??

3

2

2

3

2

lim.3

2

3lim.2

1

2lim.1

x

x

x

x

x

x

x

x

Prize!!!

What is the domain of the following function? You may not use a calculator. You will be disqualified if you do.

94)( 2 xxf

Shortcuts for Finding HA and for rational functions 1. If degree of numerator is < degree of denominator,

the limit is 0

2. If the degree of numerator = degree of denominator, the limit is the ratio of leading coefficients

3. If the degree of numerator > degree of denominator, the limit DNE

)(lim xfx

Examples. Evaluate limit and identify HA.

4

23

2

5

2

2

4lim.3

35lim.2

154

23lim.1

x

xx

x

x

xx

xx

x

x

x

Functions with 2 HA’s

Identify the Horizontal Asympotes. Prove using a limit.

12

232

x

xy

2,0 xxxFor 2,0 xxxFor

Evaluate

a.)

b.)

a.)

b.)

1

2lim

1

2lim

2

2

x

x

x

x

x

x

2

13lim

2

13lim

x

x

x

x

x

x

Use a graph or table to evaluate. 1.

2.

3.

x

x

x

x

x

x

x

sinlim

coslim

sinlim

Examples.

1.

2.

3.

xx

x

x

x

xx

x

x

x

22

sinlim

)1

cos(lim

sin5lim

Infinite Limits as x a

If the values of a function outgrow all positive bounds as x approaches a finite number a, then

If the values of a function outgrow all negative bounds as x approaches a finite number a, then

)(lim xfax

)(lim xfax

Vertical Asymptote

The line x = a is a vertical asymptote of the graph of a function y=f(x) if either

OR

)(lim xfax

)(lim xfax

Find the vertical asymptotes (if any) of the graph of the function. Prove using a limit. 1.

2.

3.

4.

)1(

2)(

16

4)(

25

158)(

1)(

2

2

2

2

2

xx

xxf

x

xxf

x

xxxf

xxf

Find the limit!!!!!(pick #’s very close to a)

1.

2.

3.

x

xx

xx

x

x

x

x

x

cos

2lim

6

32lim

2

3lim

2

2

2

3

2