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Limits at Infinity
Horizontal Asymptotes
Calculus 3.5
Limits at infinity
• “end behavior” on an infinite interval
• What does the graph do as x
approaches infinity or negative infinity?
Calculus 3.5
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Examples
• Evaluate the following limits
3x 2
lim 2
x  x  1
3x 2
lim 2
x  x  1
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Limits at Infinity
• See page 197
Calculus 3.5
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Horizontal Asymptotes
• Occur when the limit as x approaches
infinity or negative infinity exists and is
equal to L
• Limit doesn’t increase or decrease
without bound
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Theorem 3.10
• If r is a positive rational number and c is
any real number, then
c
0
r
x  x
lim
• Furthermore, if xr is defined when x < 0,
then
c
lim
0
x  x r
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Examples
• Find the limits
2 

lim  5  2 
x 
x 

2x 1
lim
x  x  1
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Indeterminate form


• Not enough information to know the limit
• Must do more work before taking the
limit
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Examples
• Find each of the limits
2x  5
lim 2
x  3 x  1
2 x2  5
lim 2
x  3 x  1
2 x3  5
lim 2
x  3 x  1
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Guidelines
• Page 200
• If degree of numerator < degree of
denominator, then L = 0
• If degree of num = degree of denom,
then L = ratio of leading coefficients
• If degree of num > degree of denom,
then L does not exist
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Examples
• Find each of the limits
lim
x 
lim
x 
3x  2
2 x2  1
3x  2
2 x2  1
Calculus 3.5
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