2.2 limits involving infinity
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Photo by Vickie Kelly, 2006. Greg Kelly, Hanford High School, Richland, Washington. 2.2 Limits Involving Infinity. North Dakota Sunset. As the denominator gets larger, the value of the fraction gets smaller. There is a horizontal asymptote if:. or. - PowerPoint PPT PresentationTRANSCRIPT
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2.2 Limits Involving Infinity
Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006
North Dakota Sunset
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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
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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.
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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
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Example 3: 5 sinlimx
x x
x
Find:
5 sinlimx
x x
x x
sinlim 5 limx x
x
x
5 0
5
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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.
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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
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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 1x
f x
g x
a left end behavior model for f if and only if
lim 1x
f x
g x
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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.
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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:
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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
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Often you can just “think through” limits.
1lim sinx x
0
0lim sinx
x
0