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[parent] asymptote (Definition)

If a plane curve $ \gamma$ has a branch continuing infinitely far from the origin $ O$, then $ \gamma$ may have an asymptote: The direct line $ l$ is an asymptote of $ \gamma$, if

$\displaystyle \lim_{d(P, \,O) \to \infty}d(P, \,l) = 0,$
where $ d(P, \,O)$ means the distance of the point $ P$ of the branch from the origin and $ d(P, \,l)$ the distance of $ P$ from the line $ l$.

Examples: The hyperbola $ \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1$ has the asymptotes $ y = \pm\frac{b}{a}x$; the curve $ y = \frac{\sin x}{x}$ the asymptote $ y = 0$.

\includegraphics{asympt1}



"asymptote" is owned by pahio. [ full author list (2) ]
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See Also: sinc function, famous curves


This object's parent.

Attachments:
asymptotes of graph of rational function (Result) by eshyvari
asymptote of Lamé's cubic (Example) by pahio
asymptote of an arctangent (Example) by PrimeFan
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Cross-references: curve, hyperbola, point, line, origin, plane curve
There are 14 references to this entry.

This is version 9 of asymptote, born on 2004-08-11, modified 2008-03-14.
Object id is 6100, canonical name is Asymptote.
Accessed 4697 times total.

Classification:
AMS MSC: 51N99 (Geometry :: Analytic and descriptive geometry :: Miscellaneous)
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