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2.2 Limits Involving Infinity
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006North Dakota Sunset
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-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 1 2 3 4
1
f x x
1lim 0 x
x
As the denominator gets larger, the value of the fraction
gets smaller.
There is a horizontal asymptote if:
lim x
f x b or lim x
f x b
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2lim 1 x
x
x
Example 1:
2lim x
x
x
This number becomes insignificant as . x
lim x
x x 1
There is a horizontal asymptote at 1.
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-2-1
0
1
2
-12 -10 -8 -6 -4 -2 2 4 6 8 10 12
sin x
f x x
Example 2: sinlim x
x x
Find:
When we graph this
function, the limit appearsto be zero.1 sin 1 x
so for :0 x 1 sin 1 x
x x x
1 sin 1lim lim lim x x x
x x x x
sin
0 lim 0 x
x x
by the sandwichtheorem:
sinlim 0 x
x
x
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Example 3: 5 sinlim x
x x x
Find:
5 sinlim x
x x x x
sinlim 5 lim x x
x x
5 0
5
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Example 4:
20
1lim
x x
20
1lim
x x
The denominator is positivein both cases, so the limit isthe same.
20
1 lim
x x
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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 1 x
f x g x
a left end behavior model for f if and only if lim 1 x
f x
g x
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Test of
modelOur modelis correct.
x f x x e Example 7:
As , approaches zero. x xe (The x term dominates.)
g x x becomes a right-end behavior model.
lim
x
x
x e x lim1
x
x
e x 1 0
1
xh x e becomes a left-end behavior model.
lim x
x x
x e
e
lim 1 x x
xe
0 1 1
As , increases faster than x decreases, x xetherefore is dominant. xe
Test ofmodel
Our modelis correct.
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x f 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 10 x 1 9 y
Use:
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5 4 2
2
2 1
3 5 7
x x x f x
x x
Example 7:
Right-end behavior models give us:
5
223
x
x
3
23
x
dominant terms in numerator and denominator
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