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Asymptotes to $y = x + \dfrac 1 x$ (blue): $y$-axis, and the line $x = y$ (red)

Let $C$ be a curve in the plane.

Let $L$ be a straight (infinite) line such that the distance between $C$ and $L$ approaches zero as they tend to infinity.

Then $L$ is an asymptote to (or of) $C$.

Also see

Linguistic Note

The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means not falling together.

It was coined by Apollonius of Perga during his analysis of conic sections.