3.5 limits at infinity determine limits at infinity determine the horizontal asymptotes, if any,...
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3.5 Limits at Infinity Determine limits at infinity
Determine the horizontal asymptotes, if any, of the graph of function.
Standard 4.5a
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Do Now: Complete the table.
x -∞ -100
-10 -1 0 1 10 100 ∞
f(x)
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x -∞ -100
-10 -1 0 1 10 100 ∞
f(x) 2 1.99
1.96
.667
0 .667
1.96
1.99
2
x decreases x increases
f(x) approaches 2 f(x) approaches 2
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Limit at negative infinity
Limit at positive infinity
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We want to investigate what happens when functions go
To Infinity and
Beyond…
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Definition of a Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if
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Limits at InfinityIf r is a positive rational number and c is any real number, then
Furthermore, if xr is defined when x < 0, then
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Finding Limits at Infinity
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Finding Limits at Infinity
is an indeterminate form
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Divide numerator and denominator by highest degree of x
Simplify
Take limits of numerator and denominator
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Guidelines for Finding Limits at
± ∞ of Rational Functions
1. If the degree of the numerator is < the degree of the denominator, then the limit is 0.
2. If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients.
3. If the degree of the numerator is > the degree of the denominator, then the limit does not exist.
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For x < 0, you can write
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Limits Involving Trig Functions
As x approaches ∞, sin x oscillates between -1 and 1. The limit does not exist.
By the Squeeze Theorem
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Sketch the graph of the equation using extrema, intercepts, and asymptotes.