1 3 Continuity End Behavior and Limits Limits

1. 3 – Continuity, End Behavior, and Limits

Limits: - Approaching a value without necessarily ever reaching it.

Non-removable Discontinuities Infinite Discontinuity: when the limit (y-value) of the function increases or decreases without bound as x approaches c from the left and right.

Non-removable Discontinuities Jump Discontinuity: the limits (y-values) of the function as x approaches c from the right and left approach two DIFFERENT values.

Removable Discontinuities the function is continuous everywhere except for at x = c.

***Note: - A limit can exist even if the value of the function at c is undefined. - The limit does not have to be the same as value of the function at c.

Testing for Continuity: A function f(x) is continuous at x = c if it satisfies ALL the following 3 conditions: 1) f(x) is defined at c. So f(c) exists. 2) f(x) approaches the same value from both sides of c. So exists.

Testing for Continuity: 3) The value that f(x) approaches from both sides of c is f(c). So.

Example 1: Determine whether continuous at continuity test. 1. Does exist? is. Justify using the

Example 1: 2. Does 3. Does exist ? ?

Example 2: Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

Example 3: Determine whether the function is continuous at x = 2. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

Example 4:

Intermediate Value Theorem

Intermediate Value Theorem Real-life example: When Bobby turned 6, he was 3 ft tall. When he turned 7, he was 3 1/2 ft tall. The IVT says that there had to be an age in between 6 and 7 that he was 3. 25 ft. He couldn’t skip any height in between 3 ft and 3 1/2 ft.