Definition:Asymptote

Definition

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

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

Examples

Example: $x^2 + \frac 1 x$

Consider the curve whose equation in the Cartesian plane is given by:

$y = x^2 + \dfrac 1 x$

This has $2$ asymptotes:

the straight line $x = 0$

the parabola $y = x^2$.

Also see

• Definition:Horizontal Asymptote
• Definition:Vertical Asymptote

• Definition:Asymptote of Hyperbola

• Results about asymptotes can be found here.

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.

The adjectival form of asymptote is asymptotic.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): asymptote
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): asymptote