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What is a horizontal asymptote?

Back

A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) approaches positive or negative infinity.

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How do you find the horizontal asymptote of a rational function?

Back

For a rational function, the horizontal asymptote can be found by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the asymptote is y = (leading coefficient of numerator)/(leading coefficient of denominator). If the degree of the numerator is greater, there is no horizontal asymptote.

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What is a vertical asymptote?

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A vertical asymptote is a vertical line that the graph of a function approaches as the input (x) approaches a certain value, typically where the function is undefined.

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How do you find the vertical asymptote of a rational function?

Back

To find the vertical asymptote of a rational function, set the denominator equal to zero and solve for x. The values of x that make the denominator zero are the vertical asymptotes.

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What does it mean if a function has a horizontal asymptote at y = -5?

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It means that as x approaches positive or negative infinity, the value of the function approaches -5.

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What does it mean if a function has a vertical asymptote at x = 3?

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It means that as x approaches 3 from the left or right, the value of the function approaches infinity or negative infinity.

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What is the significance of asymptotes in graphing functions?

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Asymptotes help to understand the behavior of a function at extreme values and where the function is undefined, guiding the overall shape of the graph.

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Can a function have more than one vertical asymptote?

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Yes, a function can have multiple vertical asymptotes, depending on the number of values that make the denominator zero.

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What is the relationship between asymptotes and limits?

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Asymptotes are closely related to limits; a function approaches an asymptote as the limit of the function approaches a certain value.

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What is the difference between a removable and non-removable discontinuity?

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A removable discontinuity occurs when a function is undefined at a point but can be defined to make it continuous. A non-removable discontinuity occurs at vertical asymptotes where the function cannot be defined.