2. MAT 125 – Applied Calculus 2.5 – One-Sided Limits & Continuity

3. Today’s Class  We will be learning the following concepts today:  One-Sided Limits  Properties of Continuous Functions  The Intermediate Value Theorem Dr. Erickson 2.5 – One-Sided Limits & Continuity 2

4. One-Sided Limits Dr. Erickson 2.5 – One-Sided Limits & Continuity 3

5. One-Sided Limits Dr. Erickson 2.5 – One-Sided Limits & Continuity 4

6. When does a limit exist? 5 2.5 – One-Sided Limits & Continuity Dr. Erickson

7. L c A general limit exists on f (x) when x = c if the right hand and left hand limits are both equal. 6 2.5 – One-Sided Limits & Continuity Dr. Erickson

8. Theorem 3 Dr. Erickson 2.5 – One-Sided Limits & Continuity 7

9. Dr. Erickson 2.5 – One-Sided Limits & Continuity 8

10. Example 1 – True or False Dr. Erickson 2.5 – One-Sided Limits & Continuity 9

11. Example 2  Find the indicated one-sided limit if it exists. Dr. Erickson 2.5 – One-Sided Limits & Continuity 10

12. Continuous Functions  Loosely speaking, a function is continuous at a point if the graph of the function at that point is devoid of holes, gaps, jumps, or breaks. Dr. Erickson 2.5 – One-Sided Limits & Continuity 11

13. Continuity of a Function at a Number Dr. Erickson 2.5 – One-Sided Limits & Continuity 12

14. Properties of Continuous Functions Dr. Erickson 2.5 – One-Sided Limits & Continuity 13

15. Properties of Continuous Functions Dr. Erickson 2.5 – One-Sided Limits & Continuity 14

16. Example 3  Find the values for x for which each function is continuous. Dr. Erickson 2.5 – One-Sided Limits & Continuity 15

17. Example 4  Find the values for x at which the function is discontinuous. Dr. Erickson 2.5 – One-Sided Limits & Continuity 16

18. Example 5 Dr. Erickson 2.5 – One-Sided Limits & Continuity 17

19. Example 6 Dr. Erickson 2.5 – One-Sided Limits & Continuity 18

20. Example 7 Dr. Erickson 2.5 – One-Sided Limits & Continuity 19

21. Theorem 4 – The Intermediate Value Theorem Dr. Erickson 2.5 – One-Sided Limits & Continuity 20

22. Example 8  Use the Intermediate Value Theorem to show that there exists a number c in the given interval such that f (c) = M. Dr. Erickson 2.5 – One-Sided Limits & Continuity 21

23. Example 9 Dr. Erickson 2.5 – One-Sided Limits & Continuity 22

24. Next Class  We will discuss the following concepts:  Slope of a Tangent Line  The Derivative  Differentiability & Continuity  Please read through Section 2.6 – The Derivative in your text book before next class. Dr. Erickson 2.5 – One-Sided Limits & Continuity 23