Warm-Up: Use the graph of f (x) to find the domain and range of the function.

1.3 – Continuity, End Behavior, and Limits

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

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. (Nonremovable Discontinuity)

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

Discontinuities Removable Discontinuity: the function is continuous everywhere except for a hole 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 is continuous at . Justify using the continuity test. 1. Does exist?

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

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: Determine whether the function 𝒇 𝒙 = 𝟓𝒙+𝟒 𝒊𝒇 𝒙>𝟐 𝟐−𝒙 𝒊𝒇 𝒙≤𝟐 is continuous at x = 2. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable.

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.

Example 5: Determine between which consecutive integers the real zeros of are located on the interval [–2, 2]. Can use graph or table.

End Behavior: Left-End Behavior Right-End Behavior

Example 6: Use the graph of f(x) = x 3 – x 2 – 4x + 4 to describe its end behavior.

Example 7: Use the graph of to describe its end behavior.