Asymptote

From Wikipedia, the free encyclopedia

For other uses, see Asymptote (disambiguation).

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.

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.

[edit] 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 horizontal 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