Limits and Continuity.

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Presentation transcript:

Limits and Continuity

Limits and Continuity Let f be a function. The f is continuous at x = a if f(a) is defined This means a is in the domain of f. This means there exists a finite limit at x = a. This means at x = a the limit is equal to the functional value. If a function is not continuous at a we say f is discontinuous at a. A function is continuous on its domain if it is continuous at each point in the domain of f. Changes: 1. Deleted period after defined to be consistent with the rest. I could not add period to the sentences that start with lim that is why I just deleted the one. Not continuous on its domain Continuous on its domain

Examples of discontinuities Limits and Continuity Examples of discontinuities f(0) is not defined. The function is continuous on its domain, but not continuous at every real number. The function does not have a limit at x = a. The limit and the functional value are not equal x = a. Changes: 1. “Examples of discontinuities” to green and moved it down a tad because it looks cut out when I play the slide show.

Your turn Limits and Continuity Given a graph, is f continuous on it domain? If not, state why. Changes: 1. Changed “your turn” to green.

Your turn Limits and Continuity Given a graph, is f continuous on it domain? If not, state why. Changes: 1. Changed “your turn” to green. Yes. 0 is not in the domain of g. Yes. 2 is not in the domain of k.

Your turn Limits and Continuity Determine what, if any, value to assign to f(a) to make f continuous at x = a. Changes: 1. Changed “your turn” to green. 2. Added “f” in “to make continuous” because it did not sound right.

Your turn Limits and Continuity Determine what, if any, value to assign to f(a) to make f continuous at x = a. Changes: 1. Changed “your turn” to green. 2. Added “f” in “to make continuous” because it did not sound right.

Limits and Continuity Use a graph to determine whether f is continuous on its domain. If it is not, list the point(s) of discontinuity. No changes. Continuous on its domain. Notice that 1 is not in the domain of f. Discontinuous at 1.