Asymptote - Definition and Overview

See also Asymptotic analysis but contrast asymptotic curve.

An asymptote to a curve is a straight line that the curve approaches in such a manner that it becomes as close as one might wish to the line by going far enough along the line. A "real-life" example of such an asymptotic relationship would involve a kitten standing 1m from a box; and, if every hour the kitten walks halfway to the box; this results in a situation in which the kitten never reaches the box; because, the distance it travels, during each hour, is never more than halfway to the box. (Compare Zeno's paradoxes.)

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. The curve approaches them, but, never reaches them. 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", although infinity is not technically considered a limit. A curve approaching a horizontal asymptote (such as in the preceding example's x = 0, which has a slope of 0) does approach a "true limit".

Asymptotes do not need to be parallel to the x- and y-axes, 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 axes, it is called an oblique asymptote.

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.

Horizontal Asymptotes Locator Theorem for Rational functions

Ask: What is the highest power of x in the numerator (N)?

What is the highest power of x in the denominator (D)?

• If N = D, then you have the 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 your Asymptote.