3.5 Limits at Infinity Determine limits at infinity Determine the horizontal asymptotes, if any, of the graph of function. Standard 4.5a
Feb 22, 2016
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 Functions1. 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 FunctionsAs 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.