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Continuity and One Sided Limits

Section 2.4. Continuity and One Sided Limits. 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. Breaking Continuity. 1. The function is undefined at x = c. Breaking Continuity. 2.

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Continuity and One Sided Limits

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

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