# Asymptote (geometry)

 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

The horizontal asymptote can be found in the same method as vertical asymptotes, but in relation to $y$ instead of $x$.

### Example Problem

Find the horizontal asymptote of $xy=1$.

### =Solution

First, we divide by $y$:

$x=\frac{1}{y}$

Clearly, the asymptote is $y=0$.

## 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.

### Example Problem

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

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

$\begin{eqnarray*}x^2&=&0\\x&=&\boxed{0\end{eqnarray*}$ (Error compiling LaTeX. ! Missing \endgroup inserted.)

So the vertical asymptote is $x=0$, or just the y-axis