Editing Asymptote (section)

== General definition ==
[[File:Graph of sect csct.svg|right|thumb|200px|(sec(t), cosec(t)), or x<sup>2</sup> + y<sup>2</sup> = (xy)<sup>2</sup>, with 2 horizontal and 2 vertical asymptotes]]

Let {{nowrap|''A'' : (''a'',''b'') &rarr; '''R'''<sup>2</sup>}} be a [[parametric curve|parametric]] plane curve, in coordinates ''A''(''t'')&nbsp;=&nbsp;(''x''(''t''),''y''(''t'')).  Suppose that the curve tends to infinity, that is:
:<math>\lim_{t\rightarrow b}(x^2(t)+y^2(t))=\infty.</math>
A line ℓ is an asymptote of ''A'' if the distance from the point ''A''(''t'') to ℓ tends to zero as ''t''&nbsp;→&nbsp;''b''.<ref>{{Citation | last1=Pogorelov | first1=A. V. | title=Differential geometry | publisher=P. Noordhoff N. V. | location=Groningen | series=Translated from the first Russian ed. by L. F. Boron | mr=0114163 | year=1959}}, §8.</ref> From the definition, only  open  curves  that  have  some  infinite branch can have an asymptote. No closed curve can have an asymptote.

For example, the upper right branch of the curve ''y''&nbsp;=&nbsp;1/''x'' can be defined parametrically as ''x''&nbsp;=&nbsp;''t'', ''y''&nbsp;=&nbsp;1/''t'' (where ''t'' > 0). First, ''x''&nbsp;→&nbsp;∞ as ''t''&nbsp;→&nbsp;∞ and the distance from the curve to the ''x''-axis is 1/''t'' which approaches 0 as ''t''&nbsp;→&nbsp;∞. Therefore, the ''x''-axis is an asymptote of the curve. Also, ''y''&nbsp;→&nbsp;∞ as ''t''&nbsp;→&nbsp;0 from the right, and the distance between the curve and the ''y''-axis is ''t'' which approaches 0 as ''t''&nbsp;→&nbsp;0. So the ''y''-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.

Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is <math>ax+by+c=0</math> then the distance from the point ''A''(''t'')&nbsp;=&nbsp;(''x''(''t''),''y''(''t'')) to the line is given by
:<math>\frac{|ax(t)+by(t)+c|}{\sqrt{a^2+b^2}}</math>
if γ(''t'') is a change of parameterization then the distance becomes
:<math>\frac{|ax(\gamma(t))+by(\gamma(t))+c|}{\sqrt{a^2+b^2}}</math>
which tends to zero simultaneously as the previous expression.

An important case is when the curve is the [[Graph of a function|graph]] of a [[real function]] (a function of one real variable and returning real values). The graph of the function ''y''&nbsp;=&nbsp;''ƒ''(''x'') is the set of points of the plane with coordinates (''x'',''ƒ''(''x'')). For this, a parameterization is
:<math>t\mapsto (t,f(t)).</math>
This parameterization is to be considered over the open intervals (''a'',''b''), where ''a'' can be &minus;∞ and ''b'' can be +∞.

An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is ''x''&nbsp;=&nbsp;''c'', for some real number ''c''. The non-vertical case has equation {{nowrap|1=''y'' = ''mx'' + ''n''}}, where ''m'' and <math>n</math> are real numbers. All three types of asymptotes can be present at the same time in specific examples.  Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.