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:

- 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.