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'asymptote'
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| Title of object: |
asymptote |
| Canonical Name: |
Asymptote |
| Type: |
Definition |
| Created on: |
2004-08-11 15:44:15 |
| Modified on: |
2005-02-19 14:32:33 |
| Classification: |
msc:51N99 |
Revision comment (for changes between this and next version):
| reuploded the pic due to minor glitch |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx} |
Content:
If a plane curve $\gamma$ has a \PMlinkescapetext{branch} continuing infinitely far from the origin $O$, then $\gamma$ may have an {\em asymptote}: \,The direct line $l$ is an asymptote of $\gamma$, if
$$\lim_{d(P, \,O) \to \infty}d(P, \,l) = 0,$$
where $d(P, \,O)$ means the \PMlinkescapetext{distance} of the point $P$ of the \PMlinkescapetext{branch} from the origin and
$d(P, \,l)$ the \PMlinkescapetext{distance} of $P$ from the line $l$.
\textbf{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 \,$y = 0$.
\begin{center}
\includegraphics{asympt1}
\end{center} |
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