Difference between revisions of "Asymptote (geometry)"

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Asymptotes can be of three different kinds: horizontal, vertical or slanted (oblique).  
 
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 <math>y</math> instead of <math>x</math>.
 
 
===Example Problem===
 
Find the horizontal asymptote of <math>xy=1</math>.
 
====Solution===
 
First, we divide by <math>y</math>:
 
 
<math>x=\frac{1}{y}</math>
 
 
Clearly, the asymptote is <math>y=0</math>.
 
  
 
== Vertical Asymptotes ==
 
== Vertical Asymptotes ==
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Find the vertical asymptotes of <math>\frac{1}{x^{2}}</math>.
 
Find the vertical asymptotes of <math>\frac{1}{x^{2}}</math>.
  
 +
====Solution====
 
To find the vertical asymptotes, <math>x^2</math> must equal zero. Solving the equation:
 
To find the vertical asymptotes, <math>x^2</math> must equal zero. Solving the equation:
  
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So the vertical asymptote is <math>x=0</math>, or just the y-axis
 
So the vertical asymptote is <math>x=0</math>, or just the y-axis
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== Horizontal Asymptotes ==
 +
The horizontal asymptote can be found in the same method as vertical asymptotes, but in relation to <math>y</math> instead of <math>x</math>.
 +
 +
===Example Problem===
 +
Find the horizontal asymptote of <math>xy=1</math>.
 +
====Solution====
 +
First, we divide by <math>y</math>:
 +
 +
<math>x=\frac{1}{y}</math>
 +
 +
Clearly, the asymptote is <math>y=0</math>.
  
 
== Slanted Asymptotes ==
 
== Slanted Asymptotes ==

Revision as of 21:15, 8 November 2007

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).


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 $\frac{1}{x^{2}}$.

Solution

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

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 $y=0$.

Slanted Asymptotes

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