Pronunciation: /ˈæ sɪmˌtoʊt/ ?

Graph of y=1/(x-1)+1 showing an asymptote at x=1
Figure 1: Graph of y=1/(x-1)+1

An asymptote is a linear boundary that a curve gets very close to, but never reaches.[1] Figure 1 is the graph of the function f(x) = 1/x. The vertical asymptote is the line x = 0. Notice that f(x) is undefined at x = 0. This means that f(x) gets very close to 0, but never reaches 0. So x = 0 is an asymptote of f(x) = 1/x.

f(x) = 1/x also has a horizontal asymptote. f(x) can never be 1 because there is no value of x such that 1/x is defined. The horizontal asymptote of f(x) = 1/x is f(x) = 1.

check mark Understanding Check

Graph of f(x) = 1/(x+1)
Figure 2: Graph of f(x)=1/(x+1)

Figure 2 is a graph of the function f(x)=1/(x+1) What are the asymptotes of this function?

Click here to see the answer.


Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1: Graph of f(x)=1/(x-x1)+y1. Created with GeoGebra
Click on the points on the slider bars in the lower right corner of the manipulative. As you slide the bars, the equation for the function changes. This also changes the asymptotes. What is the relationship between the function and the asymptotes?


  1. asymptote. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Author unknown. The Elements of the Conic Sections with the Sections of the Conoids, 3rd edition, pg 43. Published by J. Deighton and Sons, 1826. (Accessed: 2010-01-07).

Cite this article as:

Asymptote. 2010-01-07. All Math Words Encyclopedia. Life is a Story Problem LLC.


Image Credits

Revision History

2010-01-07: Added "References" (McAdams, David.)
2008-10-06: Changed understanding check so image changes on click for answer (McAdams, David.)
2008-06-25: Added understanding check and discovery (McAdams, David.)
2008-03-25: Replaced graphic in figure 1 (McAdams, David.)
2007-07-25: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License