Asymptotes

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What is an asymptote?

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An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or oblique.

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What is a vertical asymptote?

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A vertical asymptote occurs when the function approaches infinity as the input approaches a certain value. It is found by setting the denominator of a rational function to zero.

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What is a horizontal asymptote?

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A horizontal asymptote indicates the behavior of a function as the input approaches infinity. It is determined by the degrees of the numerator and denominator in a rational function.

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What is the significance of an oblique asymptote?

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An oblique (or slant) asymptote occurs when the degree of the numerator is one greater than the degree of the denominator. It represents the linear behavior of the function as x approaches infinity.

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Perform polynomial long division. The result will give the equation of the oblique asymptote.

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What is the relationship between asymptotes and limits?

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Asymptotes are closely related to limits; they describe the behavior of a function as it approaches certain values or infinity.

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What is the difference between removable and non-removable discontinuities?

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Removable discontinuities occur when a function is undefined at a point but can be defined to make it continuous. Non-removable discontinuities are associated with vertical asymptotes.

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How can you identify a removable discontinuity in a function?

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A removable discontinuity can be identified when a factor in the numerator and denominator cancels out, indicating a hole in the graph.