limits at infinity section 3.5

LIMITS AT INFINITY

Section 3.5Section 3.5

When you are done with your homework, you should

be able to…

• Determine (finite) limits at infinity• Determine the horizontal

asymptotes, if any, of the graph of a function

• Determine infinite limits at infinity 

Pythagoras lived in 550BC. He was the 1st person to teach that nature is governed by mathematics. Another

of his achievements was:A. The mathematics of musical harmony—2

tones harmonize when the ratio of their frequencies form a “simple fraction”.

B. He taught that atoms were in the shape of regular polyhedra.

C. He started a new religion (secret society) which greatly influenced western religion.

D. A and C.

DEFINITION OF LIMITS AT INFINITY

Let L be a real number.•  The statement means that

for each there exists an such that whenever

• The statement means that for each there exists an such that whenever

limx

f x L

O

f x L M O

.x M

limx

f x L

N O

f x L O

.x N

THEOREM: LIMITS AT INFINITY

If r is a positive rational number and c is any real number, then

If is defined when then •  

lim 0.rx

cx

rx 0,x

lim 0.rx

cx

Evaluate the following limit.

0.00.0

3

5limx x

HORIZONTAL ASYMPTOTES

The line is a horizontal asymptote of the graph of f if

y L

lim or lim .x x

f x L f x L

GUIDELINES FOR FINDING LIMITS AT OF RATIONAL FUNCTIONS

1. If the degree of the numerator is less than the degree of the denominator, then the limit of the rational function is 0.

2. If the degree of the numerator is equal to the degree of the denominator, then the limit of the rational function is the ratio of the leading coefficients.

3. If the degree of the numerator is greater than the degree of the denominator, then the finite limit of the rational function does not exist.

Find the horizontal asymptote(s) of the following function.

A. 3, -4B. 2C. y = 2D. B and CE. None of the above.

2

2

2 15 612

x xf xx x

Find the horizontal asymptote(s) of the following function.

A.

B.

C. D. No horizontal asymptotes.

5

5

4 5 76 1x xg xx

46

y

23

y

23

y

DEFINITION OF INFINITE LIMITS AT INFINITY

Let f be a function defined on the interval 1.The statement means that for each

positive number M there is a corresponding number such that

whenever2.The statement means that for

each negative number M there is a corresponding number such that

whenever

limx

f x

N O.x N

limx

f x

f x M

0N

, .a

f x M .x N

Find the horizontal asymptote(s) of the following function.

A.

B.

C. D. No horizontal asymptotes.

4 3

3

3 2 5 72 7

x x xr sx

32

y

y

32

y

52 6lim2x

x xx