3.5 Limits at Infinity Determine limits at infinity
Determine the horizontal asymptotes, if any, of the graph of function.
Standard 4.5a
Do Now: Complete the table.
x -∞ -100
-10 -1 0 1 10 100 ∞
f(x)
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
Limit at negative infinity
Limit at positive infinity
We want to investigate what happens when functions go
To Infinity and
Beyond…
Definition of a Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if
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
Finding Limits at Infinity
Finding Limits at Infinity
is an indeterminate form
Divide numerator and denominator by highest degree of x
Simplify
Take limits of numerator and denominator
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.
For x < 0, you can write
Limits Involving Trig Functions
As x approaches ∞, sin x oscillates between -1 and 1. The limit does not exist.
By the Squeeze Theorem
Sketch the graph of the equation using extrema, intercepts, and asymptotes.
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