ASYMPTOTE : Comprehensive Encyclopedia

An asymptote is a straight line or curve which a curve approaches as one moves along the curve. As one moves along the curve, the space between the curve and the asymptote tends to become smaller and smaller, and can be made as small as one could wish by going far enough along the curve.

The curve may or may not touch or cross the 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.

A curve can intersect its asymptote, even infinitely many times.

In the graph of 1/x, the x and y axes are the asymptotes.

In the graph of f(x)=x+1/x, the y-axis (x=0) and the line y=x are both asymptotes.

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

$\lim_{x \to \infty} f(x) = a \,\mbox{ or } \lim_{x \to -\infty} f(x) = a.$

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

$\lim_{x \to \infty} f(x) = a \,\mbox{ and } \lim_{x \to -\infty} f(x) = b$

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 graph of a function can have two horiztonal asymptotes.

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

$\lim_{x \to a-} f(x)=\pm\infty$
$\lim_{x \to a+} f(x)=\pm\infty$

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

$f(x) = \begin{cases} 1/x & x > 0 \\ 5 & x \le 0 \end{cases}$

As $\lim_{x \to 0+} f(x) = \infty$ , 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

$\lim_{x \to \infty} f(x)-(mx+b) = 0 \, \mbox{ or } \lim_{x \to -\infty} f(x)-(mx+b) = 0.$

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.