Limits to Infinity and Beyond

1. Limits to Infinity and Beyond Vertical and Horizontal asymptotes

2. I. Theorems: A.) B.)

3. D.) providing the root exists.

4. II. Vertical and Horizontal Asymptotes • A.) Def: The line x = a is a verticalasymptote of the graph of the function fiff • B.) Def: The line y = b is a horizontal asymptote of the graph of the function fiff

5. C.) Examples - Find the vertical and horizontal asymptotes for each of the following and describe the behavior at each vertical asymptote.

6. V.A. – None • H.A. y = 2 Why?

7. - V.A. –x = -3 • - H.A. – y = 1

8. D.) Example – Evaluate the following limit:

10. III. Sandwich Theorem GRAPHICALLY

11. B.) Example - What do you know about the sin function?

13. C.) Example -

14. IV. Limit Theorems

15. V. Patching In order to make our trigonometric limits look like A-D of II, we may need to “PATCH” the trig expression. After, we apply our limit properties and verify on our calculator. A) Examples -

21. V. Change of Variables A.) Trig Identities – Know Sum and Difference for sin and cos!!! B.) Sometimes it is helpful to substitute another variable when evaluating trig limits.

23. C.) Evaluate