1. 3.5 Limits at Infinity Determine limits at infinity
Determine the horizontal asymptotes, if any, of the graph of function.
Standard 4.5a

2. Do Now: Complete the table.x -∞
-100
-10 -1 1 10
100 ∞
f(x)

3. x decreases x increases f(x) approaches 2 f(x) approaches 2 x -∞ -1001 10
100 ∞
f(x) 2
1.99
1.96
.667
f(x) approaches 2
f(x) approaches 2

4. Limit at negative infinity
Limit at positive infinity

5. To Infinity and Beyond…
We want to investigate what happens when functions go
To Infinity and Beyond…

7. Definition of a Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if

8. Limits at Infinity
If r is a positive rational number and c is any real number, then
Furthermore, if xr is defined when x < 0, then

9. Finding Limits at Infinity

10. Finding Limits at Infinity
is an indeterminate form

11. Divide numerator and denominator by highest degree of x
Simplify
Take limits of numerator and denominator

14. Guidelines for Finding Limits at ± ∞ of Rational Functions
If the degree of the numerator is < the degree of the denominator, then the limit is 0.
If the degree of the numerator = the degree of the denominator, then the limit is the ratio of the leading coefficients.
If the degree of the numerator is > the degree of the denominator, then the limit does not exist.

15. For x < 0, you can write

16. Limits Involving Trig Functions
As x approaches ∞, sin x oscillates between -1 and 1. The limit does not exist.
By the Squeeze Theorem

17. Sketch the graph of the equation using extrema, intercepts, and asymptotes.