Asymptote

For other uses, see Asymptote (disambiguation).
"Asymptotic" redirects here. Not to be confused with Asymptomatic.
The graph of a function with a horizontal (y = 0), vertical (x = 0), and oblique asymptote (purple line, given by y = 2x).
A curve intersecting an asymptote infinitely many times.

In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.[1][2]

The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen".[3] The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.[4]

There are three kinds of asymptotes: horizontal, vertical and oblique. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.

More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.

Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.[5] The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.

  1. ^ Williamson, Benjamin (1899), "Asymptotes", An elementary treatise on the differential calculus
  2. ^ Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane", Mathematics Magazine, 72 (3): 183–192, CiteSeerX 10.1.1.502.72, doi:10.2307/2690881, JSTOR 2690881
  3. ^ Oxford English Dictionary, second edition, 1989.
  4. ^ D.E. Smith, History of Mathematics, vol 2 Dover (1958) p. 318
  5. ^ Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-00005-1, §4.18.