1. Continuity and One Sided Limits
Section 2.4
Continuity and One Sided Limits

2. Continuity
To say a function is continuous at x = c means that there is NO interruption in the graph of f at c. The graph has no holes, gaps, or jumps.

3. Breaking Continuity
1. The function is undefined at x = c

4. Breaking Continuity 2.

5. Breaking Continuity 3.

6. Definition of Continuity
A function f is continuous at c IFF ALLare true…
1. f(c) is defined. 2. 3.
A function is continuous on an interval (a, b) if it is continuous at each pt on the interval.

7. Discontinuity
A function is discontinuous at c if f is defined on (a, b) containing c (except maybe at c) and f is not continuous at c.

8. 2 Types of Discontinuity
1. Removable :
You can factor/cancel out, therefore making it continuous by redefining f(c).
2. Non-Removable:
You can’t remove it/cancel it out!

9. Examples: 1. Removable: We “removed” the (x-2).
Therefore, we have a REMOVABLE DISCONTINUITY when x – 2 = 0, or, when x = 2.

10. Non-Removable:
We can’t remove/cancel out this discontinuity, so we have a NON-Removable discontinuity when x – 1 =0, or when x = 1.
We will learn that Non-Removable Discontinuities are actually Vertical Asymptotes!

11. To Find Discontinuities…
1. Set the deno = 0 and solve.
2. If you can factor and cancel out (ie-remove it) you have a REMOVABLE Discontinuity.
3. If not, you have a NON-Removable Discontinuity.

12. One Sided Limits
You can evaluate limits for the left side, or from the right side.

13. Limits from the Right
x approaches c from values that are greater than c.

14. Limits from the Left
x approaches c from values that are less than c.

15. Find each limit… 1.
= 0

16. Therefore, the limit as x approaches 0 DNE!!2.
= 1
= -1
Therefore, the limit as x approaches 0 DNE!!

17. 3. 3 3 3

18. Steps for Solving One Sided Limits
1. Factor and cancel as usual.
2. Evaluate the resulting function for the value when x=c.
3. If this answer is NOT UNDEFINED then that is your solution.
4. If this answer is UNDEFINED, then graph the function and look at the graph for when x=c.

19. Steps for Solving a Step Function
Ex:
Evaluate each function separately for the value when x=c.
If the solutions are all the same, that is your limit.
If they are not, then the limit DNE.

20. Therefore, the limit as x approaches 1 of f(x) =1
= 1
Therefore, the limit as x approaches 1 of f(x) =1
= 1