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

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  $y=0$.