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2.2 Limits Involving Infinity
North Dakota Sunset
1f x
x
1lim 0x x
As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if:
limx
f x b
or limx
f x b
2lim
1x
x
x
Example 1:
2limx
x
x
This number becomes insignificant as .x
limx
x
x 1
There is a horizontal asymptote at 1.
sin xf x
x
Example 2:
sinlimx
x
x Find:
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
Infinite Limits:
1f x
x
0
1limx x
As the denominator approaches zero, the value of the fraction gets very large.
If the denominator is positive then the fraction is positive.
0
1limx x
If the denominator is negative then the fraction is negative.
vertical asymptote at x=0.
Example 4:
20
1limx x
20
1limx x
The denominator is positive in both cases, so the limit is the same.
20
1 limx x
End Behavior Models:
End behavior models model the behavior of a function as x approaches infinity or negative infinity.
A function g is:
a right end behavior model for f if and only if
lim 1x
f x
g x
a left end behavior model for f if and only if
lim 1x
f x
g x
Test ofmodel
Our modelis correct.
xf x x e Example 7:
As , approaches zero.x xe(The x term dominates.)
g x x becomes a right-end behavior model.
limx
x
x e
x
lim1
x
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 , increases faster than x decreases,x xe
therefore is dominant.xe
Test ofmodel
Our modelis correct.
xf x x e Example 7:
g x x becomes a right-end behavior model.
xh x e becomes a left-end behavior model.
On your calculator, graph:
1
2
3
x
x
y x
y e
y x e
10 10x
1 9y
Use:
5 4 2
2
2 1
3 5 7
x x xf x
x x
Example 7:
Right-end behavior models give us:
5
2
2
3
x
x
32
3
x
dominant terms in numerator and denominator
Often you can just “think through” limits.
1lim sinx x
0
0lim sinx
x
0
p
