3.5 Limits at Infinity

1. 3.5 Limits at Infinity "The only angle from which to approach a problem is the TRY-Angle"

2. Objective • To evaluate limits as a function approaches infinity

3. A change in approach • We’ve been looking at limits as they appraoch a certain value • x  7, x  -3, x  2,135 • We can also look as x approaches infinity or negative infinity

4. Example by chart

5. Or by graph

6. Thm: Limits at Infinity • If r is a positive rational number and c is a number then • Also, if x^r is defined when x<0, then

7. Example

8. Rewriting

9. 3 Rules • 1. If the numerator’s exponent is greater than the denominator then the limit is infinity or negative infinity • 2. If the numerator’s exponent is less than the denominator then the limit is zero • 3. If the numerator’s exponent is equal to the denominator then the limit is the ratio of coefficients

10. Examples

11. Trig functions

12. Proof:

13. Horizontal asymptotes • The line y = L is a h.a. of the graph if:

14. In other words… • To find h.a. take the limit as a function approaches infinity and/or negative infinity