Continuity and One Sided Limits

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Presentation transcript:

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.

Breaking Continuity 3.

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.

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.

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!

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

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!

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.

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

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

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

Find each limit… 1. = 0

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

3. 3 3 3

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.

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.

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