Editing Asymptote (section)

{{Short description|Limit of the tangent line at a point that tends to infinity}}
{{Other uses}}
{{Redirect-distinguish|Asymptotic|Asymptomatic}}
[[File:Asymptotic curve hvo1.svg|right|thumb|250px|The graph of a function with a horizontal (''y''&nbsp;=&nbsp;0), vertical (''x''&nbsp;=&nbsp;0), and oblique asymptote (purple line, given by ''y''&nbsp;=&nbsp;2''x'')]]
[[File:Asymptote02 vectorial.svg|right|thumb|250px|A curve intersecting an asymptote infinitely many times]]

In [[analytic geometry]], an '''asymptote''' ({{IPAc-en|ˈ|æ|s|ɪ|m|p|t|oʊ|t|audio=LL-Q1860 (eng)-Naomi Persephone Amethyst (NaomiAmethyst)-asymptote.wav}}) 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 [[Limit of a function#Limits at infinity|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]].<ref>{{citation|title=An elementary treatise on the differential calculus|chapter=Asymptotes|first=Benjamin|last=Williamson|chapter-url=https://books.google.com/books?id=znsXAAAAYAAJ&pg=241|year=1899}}</ref><ref>{{citation|first=Jeffrey|last=Nunemacher|title=Asymptotes, Cubic Curves, and the Projective Plane|journal=Mathematics Magazine|volume=72|issue=3|year=1999|pages=183–192|jstor=2690881|doi=10.2307/2690881|citeseerx=10.1.1.502.72}}</ref>

The word asymptote is derived from the [[Greek language|Greek]] ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ [[Privative alpha|priv.]] + σύν "together" + πτωτ-ός "fallen".<ref>''Oxford English Dictionary'', second edition, 1989.</ref> 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.<ref>D.E. Smith, ''History of Mathematics, vol 2'' Dover (1958) p. 318</ref>

There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the [[graph of a function|graph]] of a [[function (mathematics)|function]] {{nowrap|1=''y'' = ''&fnof;''(''x'')}}, horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to {{nowrap|+&infin; or &minus;&infin;.}}  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 {{nowrap|+&infin; or &minus;&infin;.}}

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.<ref>{{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra | publisher=[[John Wiley & Sons]] | location=New York | edition=2nd | isbn=978-0-471-00005-1 | year=1967 | url-access=registration | url=https://archive.org/details/calculus01apos }}, §4.18.</ref>  The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of [[asymptotic analysis]].