Asymptote

An asymptote is a straight line which accompanies a curve at an arbitrary distance.

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A specific example of asymptotes can be found in the graph of the function f(x) = 1/x, in which two asymptotes are being approached: the line y = 0 and the line x = 0. A curve approaching a vertical asymptote (such as in the preceding example's y = 0, which has an undefined slope) could be said to approach an "infinite limit".

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Asymptotes need not be parallel to the x- or y-axis, as shown by the graph of x + x−1, which is asymptotic to both the y-axis and the line y = x. When an asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote.

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A function f(x) can be said to be asymptotic to a function g(x) as x → ∞. This has any of four distinct meanings:

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• f(x) − g(x) → 0.
• f(x) / g(x) → 1.
• f(x) / g(x) has a nonzero limit.
• f(x) / g(x) is bounded and does not approach zero. See Big O notation.

:See also asymptotic analysis, but contrast with asymptotic curve.

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Asymptote: Horizontal asymptotes locator theorem for rational functions ►



 

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