3.5 Limits at Infinity

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.

3. x decreases x increases f(x) approaches 2 f(x) approaches 2

4. Limit at negative infinity • Limit at positive infinity

5. 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.