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
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