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    An asymptote is a straight line or curve A to which another curve B (the one being studied) approaches closer and closer as one moves along it. As one moves along B, the space between it and the asymptote A becomes smaller and smaller, and can in fact be made as small as one could wish by going far enough along. A curve may or may not touch or cross its asymptote. In fact, the curve may intersect the asymptote an infinite number of times.

    If a curve C has the curve L as an asymptote, one says that C is asymptotic to L.







        Asymptote
            Asymptotes and graphs of functions
            Other meanings

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    Asymptotes and graphs of functions

    Asymptotes are formally defined using limits.

    Suppose f is a function. Then the line y=a is a horizontal asymptote for f if
    lim_ f(x) = a ,mbox lim_ f(x) = a.

    Intuitively, this means that for large (positive, or negative) values of
    x, the value of f(x) is approximately equal to a, and that the approximation becomes
    better and better as x becomes larger (in a general sense: there can be local wobbling).
    This means that far out on the curve, the curve will be close to the line.

    Note that if
    lim_ f(x) = a ,mbox lim_ f(x) = b

    then the graph of f has two horizontal asymptotes: y=a and y=b. An example of
    such a function is the arctangent function.



    The line x=a is a vertical asymptote of a function f if either of the following conditions is true:
      lim_ f(x)=pminfty
      lim_ f(x)=pminfty

    Intuitively, if x=a is an asymptote of f, then, if we imagine x approaching a from one side, the
    value of f(x) grows without bound; i.e., f(x) becomes large (positively or negatively), and, in
    fact, becomes larger than any set value.

    A specific example of asymptotes can be found in the graph of the function f(x) = 1/x, in which two asymptotes are seen: the horizontal line y = 0 and the vertical line x = 0.

    Note that f(x) may or may not be defined at a: what the function is doing
    precisely at x=a does not affect the asymptote. For example, consider the function

    f(x) = egin 1/x & x > 0 \ 5 & x le 0 end


    As lim_ f(x) = infty, f(x) has a vertical asymptote at 0, even though f(0) = 5.

    Asymptotes of a graph of a function need not be parallel to the x- or y-axis, as shown by the graph of f(x)=x +1/x,
    which is asymptotic to 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.
    If y= m x + b, is any non-vertical line, then the function f(x) is
    asymptotic to it if
    lim_ f(x)-(mx+b) = 0 , mbox lim_ f(x)-(mx+b) = 0.

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    Other meanings

    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.
     
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    Scientus.org Dictionary (Yet Another Wiki) RC : 1.39
    This article is licensed under the GNU Free Documentation License [copyleft]. It uses material from the Wikipedia article "Asymptote". link