1. 1.4 One-Sided Limits and Continuity

2. Definition A function is continuous at c if the following three conditions are met 2. Limit of f(x) exists 1. f(c) is defined 3. Limit of f(x) is c c

3. Definition If a function is defined on an interval I, except at c, then the function is said to have a discontinuity at c such as a hole, break or asymptote

4. One-Sided Limits Approach a function from different directions both graphically and analytically 1)Limits from the right 2)Limits from the left

5. Existence of a Limit Let be a function and let c and L be real numbers. The limit of as x approaches c is L if and only if (iff)

6. Consider ( | ) a c b ( | ) a c b ( | ) a c b

7. 1) Find

8. Left Right 2) Find

9. LeftRight By existence theorem

10. 3) Determine if the limit exists at x = -2 if

11. Left Right

12. Continuity at a point Let f(x) be defined on an open interval containing c, f(x) is continuous at c if a. is defined (exists) b. exists (one-sided limits are equal) c.The

13. Discontinuity Removable: the function can be redefined (hole discontinuity)

14. Discontinuity Non - Removable: a.Jump - breaks at a particular value b. Infinite discontinuity - vertical asymptote

15. 4)Find the x - values where is not continuous and classify a. exists b. c. Removable Point Discontinuity

16. 5)Find the x - values at which is not continuous, is the discontinuity removable? Non-removable: asymptotes

17. 6)Find the x - values at which is not continuous, is the discontinuity removable? Non-removable Removable

18. 7)If is continuous at, then

19. HOMEWORK Page 79 # 1-11, 18, 19, and 20