asymptote

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

$$\underset{d(P,O)\to \mathrm{\infty }}{lim}d(P,l)=0,$$

where $d(P,O)$ means the of the point $P$ of the from the origin and$d(P,l)$ the 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  $y=0$.