Limits & Continuity Final Exam Review

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What is continuity at a point?

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A function is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit equals f(c).

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What are the types of discontinuities?

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1) Removable Discontinuity: A hole in the graph. 2) Infinite Discontinuity: A vertical asymptote. 3) Jump Discontinuity: A sudden change in value.

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What does it mean if a function has a removable discontinuity at x = c?

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It means that the limit of the function exists at x = c, but the function is not defined at that point.

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How do you determine if a limit exists as x approaches a certain value?

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A limit exists if the left-hand limit and right-hand limit as x approaches that value are equal.

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What is the definition of a function being continuous on an interval?

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A function is continuous on an interval if it is continuous at every point in that interval.

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What is the significance of the statement 'f(c) must exist' in continuity?

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It is one of the necessary conditions for a function to be continuous at x = c.

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What does it mean if a function is not continuous at x = 0?

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It means that at least one of the conditions for continuity is not satisfied at that point.

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What is a vertical asymptote?

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A vertical asymptote occurs when the function approaches infinity or negative infinity as x approaches a certain value.

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What is the difference between a limit and a function value?

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A limit describes the behavior of a function as it approaches a certain point, while the function value is the actual output of the function at that point.