Section 3.5
Limits at Infinity
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3x 2
f ( x)  2
x 1


Discuss “end behavior” of a function on an
interval
Graph:
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2
3x
lim 2
x  x  1


As x increases without bound f(x)
approaches _______.
As x decreases without bound f(x)
approaches _______.
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Definition of a Horizontal Asymptote
The line y=L is a horizontal asymptote
of the graph of f if
lim f ( x)  L

x 
OR
lim f ( x)  L
x 
Note: a function can have at most 2 horizontal
asymptotes
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Exploration




Use a graphing utility to graph
y=(2x^2 +4x-6)/(3x^2+2x-16)
Describe all important features of the
graph.
Can you find a single viewing window that
shows all these features clearly?
What are the horizontal asymptotes?
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Recall Limits at infinity:
1. If r is a positive rational number and c is any
c
lim
0
real number, then
x
Furthermore, if x^r is defined when x<0,
then lim c  0
x 
x 
r
xr
2. If the degree of the numerator is less than
the degree of the denominator, then the
limit of the rational function is 0.
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3. If the degree of the numerator is equal
the degree of the denominator, then the
limit of the rational function is the ratio of
the leading coefficients.
4. If the degree of the numerator is greater
than the degree of the denominator, then
the limit of the rational function DNE.
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Examples
2
1) lim(4  2 )
x 
x
2x  3
)
2) lim(
x  x  1
2x  5
3) lim( 2
)
x  3 x  1
3
2
x
5
4) lim(
)
2
x  3 x  1
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More on Horizontal Asymptotes


Rational functions always have the same
horizontal asymptote to the right and to
the left. Functions that are NOT rational
may approach different horizontal
asymptotes.
EX:
3x  2
3x  2
lim
x 
2 x2  1
and
lim
x 
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2 x2  1
10
The graph:
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Limits involving Trig Functions
1)
lim cos x
2)
lim
x 
sin x
x 
x
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Infinite Limits at Infinity
Examples:
1)
3
lim x
2)
x 
x 
3)
2 x2  4 x
lim
x 
x 1
lim x 3
4)
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2x2  4x
lim
x 
x 1
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Asymptotes


www.purplemath.com/modules/asymtote4
.htm
Slant asymptotes: The graph of a rational
function (having no common factors) has
a slant asymptote if the degree of the
numerator exceeds the degree of the
denominator by 1.
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Find an equation for the slant
asymptote:

Use long Division:
2x2  4x
y
x 1
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References

Larson, Hostetler, and Edwards, Calculus of a Single Variable,
Houghton Mifflin Company: 2002
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