Asymptote (geometry)

Revision as of 19:08, 8 November 2007 by Hunter34 (talk | contribs) (Vertical Asymptotes)
This is an AoPSWiki Word of the Week for Nov 8-14
For the vector graphics language, see Asymptote (Vector Graphics Language).

An asymptote is a line or curve that a certain function approaches.

Asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).

Horizontal Asymptotes

Vertical Asymptotes

The vertical asymptote can be found by finding values of $x$ that make the function undefined. One of the common ways is to have the function divided by zero, which is undefined. This can be shown by example.

Find the vertical asymptotes of $\frac{1}{x^{2}}$.

To find the vertical asymptotes, $x^2$ must equal one. Solving the equation:

$\begin{eqnarray*}x^2&=&1\\x&=&\boxed{-1,1}\end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)

So the vertical asymptote is $x=-1$ and $x=1$

Slanted Asymptotes

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