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).
Contents
Horizontal Asymptotes
The horizontal asymptote can be found in the same method as vertical asymptotes, but in relation to 
instead of 
.
Example Problem
Find the horizontal asymptote of 
.
=Solution
First, we divide by :
Clearly, the asymptote is 
.
Vertical Asymptotes
The vertical asymptote can be found by finding values of 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 
.
To find the vertical asymptotes, 
must equal zero. Solving the equation:
$\begin{eqnarray*}x^2&=&0\\x&=&\boxed{0\end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)
So the vertical asymptote is 
, or just the y-axis
Slanted Asymptotes
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