2.2 Limits Involving Infinity

Created by Greg Kelly, Hanford High School, Richland, WashingtonRevised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

1f x

x

1lim 0x x

As the denominator gets larger (as x →∞), the overall fraction value gets smaller.

There is a horizontal asymptote if either:

limx

f x b

or limx

f x b

Limits outward “toward infinity”Limits outward “toward infinity”

2lim

1x

x

x

Example 1:

2limx

x

x

Adding 1 becomes insignificant as .x

1

There is a horizontal asymptote at a height limit of 1.

||lim

x

xx

limx

x

x

sin xf x

x

Example 2:

sinlimx

x

xFind:

When we graph this function, the limit appears to be zero.

1 sin 1x

so for :0x 1 sin 1x

x x x

1 sin 1lim lim limx x x

x

x x x

sin0 lim 0

x

x

x

by the Sandwich Theorem:

sinlim 0x

x

x

Example 3: 5 sinlimx

x x

x

Find:

5 sinlimx

x x

x x

sinlim 5 limx x

x

x

5 0

5

Limits that are infinite in height

1f x

x

0

1limx x

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

If the denominator is positive, then the fraction’s overall sign is positive.

0

1limx x

If the denominator is negative, then the fraction’s overall sign is negative.

vertical asymptote at x=0.

The overall limit as x approaches zero does not exist,since the left-hand and right-hand limits don’t match.

Example 4:

The denominator is positive after squaring x, so the limit has the same sign from left and right.

20

1 limx x

20

1lim

xx

20

1lim

xx

The overall limit as x approaches zero is “positive without bound,” since the left-hand and right-hand limits both trend toward positive infinity.

End Behavior Models

An end behavior model describes the height behavior of a function as x approaches positive infinity or

as x approaches negative infinity.

A function g(x) is:

a right-end-behavior model for f if and only if

lim 1x

f

x

x

g

a left-end-behavior model for f if and only if

lim 1x

f x

g x

Test ofmodel

xf x x e Example 7:

As , approaches zero...x xe whereas x is further from the x-axis, so x dominates on the right.

xg x becomes a right-end behavior model.

limx

x

x e

x

lim 1x

x

e

x

1 0 1

xh x e becomes a left-end behavior model.

limx

xx

x e

e

lim 1xx

x

e

0 1 1

As , is much further away from the x-axis,x xe

therefore is dominant on the left.xe

Test ofmodel

Our modelis correct!

Find right- and left-end models for

xf x x e Example 7:

xg x becomes a right-end behavior model.

xeh x becomes a left-end behavior model.

On your calculator, compare which pairs of graphs match the best on the right and on the left of the y-axis:

1

2

3

x

x

y x

y e

y x e

10 10x

1 9y

Window:

5 4 2

4

2 1

3 5 7

x x xf x

x x

Example:

For rational functions, the end behavior model comes from the highest power termsin the numerator and denominator:

5

4

2

3

x

x

2

3x

dominant terms in numerator and denominator

when |x| largef(x) ≈

You can adjust the direction of limits:“outbound” limit for an input

equals

“inbound” limit (toward x = 0) for the input’s reciprocal

lim sin1

x x

0

lim sin( )x

x

0

0lim ( ) lim

1x x

f fx

x