Asymptote
From Free net encyclopedia
- For other uses, see Asymptote (disambiguation).
An asymptote is a straight or curved line which a curve will approach arbitrarily closely, but never touch.
Image:1-over-x.png 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 do not need to go to infinity when approached at both sides. Asymptote x=a is a vertical asymptote for f(x) if it just satisfies either of the following conditions:
- <math>\lim_{x \to a-} f(x)=\pm\infty</math>
- <math>\lim_{x \to a+} f(x)=\pm\infty</math>
Note that f(x) need not be undefined at a. For example, consider the function
- <math>f(x) = \begin{cases} 1/x & x \neq 0 \\ 5 & x = 0 \end{cases}</math>
As both <math>\lim_{x \to 0+} f(x) = \infty</math> and <math>\lim_{x \to 0-} f(x) = -\infty</math>, f(x) has a vertical asymptote at 0, even though <math>f(0) = 5</math>.
A function f(x) can be said to be asymptotic to a function g(x) as x → ∞. This has any of four distinct meanings:
- 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.da:Asymptote
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