Asymptote
Pronunciation: /ˈæ sɪmˌtoʊt/ ?
 Figure 1: Graph of y=1/(x1)+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.

Understanding Check
 Figure 2: Graph of f(x)=1/(x+1) 
 Figure 3: 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.
The asymptotes of
f(x)=1/(x+1)
are
x=1
and
f(x)=0

Discovery
 Manipulative 1: Graph of f(x)=1/(xx1)+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?

References
 asymptote. http://wordnet.princeton.edu/. WordNet. Princeton University. (Accessed: 20110108). http://wordnetweb.princeton.edu/perl/webwn?s=asymptote&sub=Search+WordNet&o2=&o0=1&o7=&o5=&o1=1&o6=&o4=&o3=&h=.
 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: 20100107). http://www.archive.org/stream/elementsofconics00hastuoft#page/43/mode/1up/search/asymptote.
Cite this article as:
Asymptote. 20100107. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/asymptote.html.
Translations
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Revision History
20100107: Added "References" (
McAdams, David.)
20081006: Changed understanding check so image changes on click for answer (
McAdams, David.)
20080625: Added understanding check and discovery (
McAdams, David.)
20080325: Replaced graphic in figure 1 (
McAdams, David.)
20070725: Initial version (
McAdams, David.)