Editing Asymptote (section)

==Curvilinear asymptotes==
[[File:nonlinear asymptote.svg|right|thumb|200px|''x''<sup>2</sup>+2''x''+3 is a parabolic asymptote to (''x''<sup>3</sup>+2''x''<sup>2</sup>+3''x''+4)/''x.'']]
Let {{nowrap|''A'' : (''a'',''b'') &rarr; '''R'''<sup>2</sup>}} be a parametric plane curve, in coordinates ''A''(''t'')&nbsp;=&nbsp;(''x''(''t''),''y''(''t'')), and ''B'' be another (unparameterized) curve. Suppose, as before, that the curve ''A'' tends to infinity. The curve ''B'' is a curvilinear asymptote of ''A'' if the shortest distance from the point ''A''(''t'') to a point on ''B'' tends to zero as ''t''&nbsp;→&nbsp;''b''.  Sometimes ''B'' is simply referred to as an asymptote of ''A'', when there is no risk of confusion with linear asymptotes.<ref>{{citation|last=Fowler|first=R. H.|title=The elementary differential geometry of plane curves|journal=Nature |publisher=Cambridge, University Press|year=1920|volume=105 |issue=2637 |page=321 |doi=10.1038/105321a0 |bibcode=1920Natur.105..321G |url=https://archive.org/details/elementarydiffer00fowlrich|isbn=0-486-44277-2|hdl=2027/uc1.b4073882}}, p. 89ff.</ref>

For example, the function
:<math>y = \frac{x^3+2x^2+3x+4}{x}</math>
has a curvilinear asymptote {{nowrap|1=''y'' = ''x''<sup>2</sup> + 2''x'' + 3}}, which is known as a ''parabolic asymptote'' because it is a [[parabola]] rather than a straight line.<ref>William Nicholson, ''The British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge'', Vol. 5, 1809</ref>