VERTICAL AND HORIZONTAL ASYMPTOTES Limits to Infinity and Beyond

I. Theorems: A.) B.)

D.) providing the root exists.

II. Vertical and Horizontal Asymptotes A.) Def: The line x = a is a vertical asymptote of the graph of the function f iff B.) Def: The line y = b is a horizontal asymptote of the graph of the function f iff

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

- V.A. – None - H.A. y = 2Why?

- V.A. – x = -3 - H.A. – y = 1

D.) Example – Evaluate the following limit:

III. Sandwich Theorem GRAPHICALLY

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

C.) Example -

IV. Limit Theorems

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 -

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.

C.) Evaluate