2.6 Limits at Infinity

|x

As x approaches infinity

f(x) ? as x gets larger and larger

f(x) ? as x gets larger and larger in the negative direction

8

f (x)

x → ∞

8)(lim

xfx

lim ( )x

f x

Fun Fact: A student, who is an excellent pattern-observer, was asked to find the following limit. He quickly answered

)(lim7

xfx

7

|x

Limit at infinity

If f(x) approaches L as x approaches ∞ or - ∞, then

the limit as x approaches a of f(x) is L.

L

f (x)

x → ∞

lim ( )x

f x L

means that the value of f(x) can be made as close to L as we like by taking x sufficiently large.

Graphical Approach

)(lim xfx

)(lim2

xfx

)(lim2

xfx

)2(f

0

dne undefined

Vertical Asymptote:x = -2

Horizontal Asymptote:y = 0

Horizontal Asymptote

The line y = L is called a horizontal asymptote of the curve f (x) if either

lim ( )x

f x L

lim ( )x

f x L

or

L

f

Examples

Find the limits.

xx

3lim

5

3lim

2xx

5

3lim

5xx

lim 0x

xe

Horizontal Asymptote

lim 0x

xe

Since

f(x) = ex has a horizontal asymptote y = 0

Polynomial functions have no horizontal asymptotes.

Find all horizontal asymptotes of the function.

xexf

3

5)( xxg arctan)(

Finding limits at infinity

lim 0nx

c

x If n is a positive number, then

To find limits at infinity, we multiply the expression by

where n is the highest power in the denominator.

Then use the above property to simplify.

1/

1/

n

n

x

x

Examples

Find the limits.

2

2

2lim

3 1t

t

t t

xx

xx 3

52lim

3

13

45lim

4

25

xx

xxx

2

2

2lim

3 1t

t

t t

2

2lim

9 1y

y

y

2

2

sinlimx

x

x