Asymptote

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An asymptote is a straight or curved line which a curve will approach arbitrarily closely.

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 the preceding example's x = 0, which has an undefined slope) could be said to approach an "infinite limit," while a curve approaching a horizontal line (such as the previous example's y = 0) could be said to approach a limit at infinity.

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.

Asymptotes, especially vertical asymptotes, also not need to go to infinity when approached at both sides. Asymptote x=a is a vertical asymptote for f(x) if it just satifies at least one of the following conditions:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to a-} f(x)=\infty}
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to a+} f(x)=\infty}
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to a-} f(x)=-\infty}
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to a+} f(x)=-\infty}

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

  1. f(x) − g(x) → 0.
  2. f(x) / g(x) → 1.
  3. f(x) / g(x) has a nonzero limit.
  4. f(x) / g(x) is bounded and does not approach zero. See Big O notation.
See also asymptotic analysis, but contrast with asymptotic curve.

Horizontal asymptotes locator theorem for rational functions

Ask: What is the highest power n of x in the numerator?

What is the highest power d of x in the denominator?

  • If n = d, then there is an asymptote at y = leading coefficient of numerator/leading coefficient of denominator
  • If n > d Then there is no strictly horizontal asymptote
  • If n < d Then the x-axis is a horizontal asymptote.

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