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

2. 1.3 – Continuity, End Behavior, and Limits

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

4. 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)

5. 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)

6. Discontinuities
Removable Discontinuity: the function is continuous everywhere except for a hole at x = c.

7. ***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.

8. 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.

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

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

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

12. 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.

13. 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.

14. 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.

15. Intermediate Value Theorem

16. 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.

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

18. End Behavior:
Left-End Behavior Right-End Behavior

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

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