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. 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".

An 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.

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.