asymptoteAsymptote2004-08-11 15:44:152008-03-14 19:51:19Definitioncurve\usepackage{amssymb}
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\usepackage{graphicx}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\, $y = 0$.
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