Editing Asymptote (section)

==Asymptotes of functions==
The asymptotes most commonly encountered in the study of [[calculus]] are of curves of the form {{nowrap|1=''y'' = ''&fnof;''(''x'')}}. These can be computed using [[limit (mathematics)|limits]] and classified into ''horizontal'', ''vertical'' and ''oblique'' asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to +∞ or &minus;∞. As the name indicates they are parallel to the ''x''-axis. Vertical asymptotes are vertical lines (perpendicular to the ''x''-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as ''x'' tends to +∞ or &minus;∞.

===Vertical asymptotes===
The line ''x'' = ''a'' is a ''vertical asymptote'' of the graph of the function {{nowrap|1=''y'' = ''&fnof;''(''x'')}} if at least one of the following statements is true:

# <math>\lim_{x \to a^{-}} f(x)=\pm\infty,</math>
# <math>\lim_{x \to a^{+}} f(x)=\pm\infty,</math>

where <math>\lim_{x\to a^-}</math> is the limit as ''x'' approaches the value ''a'' from the left (from lesser values), and <math>\lim_{x\to a^+}</math> is the limit as ''x'' approaches ''a'' from the right.

For example, if ƒ(''x'') = ''x''/(''x''–1), the numerator approaches 1 and the denominator approaches 0 as ''x'' approaches 1. So
:<math>\lim_{x\to 1^{+}}\frac{x}{x-1}=+\infty</math>
:<math>\lim_{x\to 1^{-}}\frac{x}{x-1}=-\infty</math>
and the curve has a vertical asymptote ''x'' = 1.

The function ''ƒ''(''x'') may or may not be defined at ''a'', and its precise value at the point ''x'' = ''a'' does not affect the asymptote.  For example, for the function

:<math>f(x) = \begin{cases} \frac{1}{x} & \text{if } x > 0, \\ 5 & \text{if  } x \le 0. \end{cases}</math>

has a limit of +∞ as {{nowrap|''x'' &rarr; 0<sup>+</sup>}}, ''ƒ''(''x'') has the vertical asymptote  {{nowrap|1=''x'' = 0}}, even though ''ƒ''(0)&nbsp;=&nbsp;5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or [[vertical line test|a vertical line in general]]) in more than one point. Moreover, if a function is [[continuous function|continuous]] at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.

A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.

If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
:<math>f(x) =  \tfrac 1x + \sin(\tfrac 1x)\quad</math> at <math>\quad x=0</math>.

This function has a vertical asymptote at <math>x=0,</math> because 
:<math>\lim_{x\to0^+} f(x) = \lim_{x\to0^+}\left(\tfrac 1x + \sin\left(\tfrac 1x\right)\right) = +\infty,</math>

and

:<math>\lim_{x\to0^-} f(x) = \lim_{x\to0^-}\left(\tfrac 1x + \sin\left(\tfrac 1x\right)\right) = -\infty</math>.

The derivative of <math>f</math> is the function
:<math>f'(x)=\frac{-(\cos(\tfrac 1x) + 1)}{x^2}</math>.

For the sequence of points
:<math>x_n=\frac{(-1)^n}{(2n+1)\pi},\quad</math> for <math>\quad n=0,1,2,\ldots</math>

that approaches  <math>x=0</math> both from the left and from the right, the values <math>f'(x_n)</math> are constantly <math>0</math>. Therefore, both [[one-sided limit]]s of <math>f'</math> at <math>0</math> can be neither <math>+\infty</math> nor <math>-\infty</math>. Hence <math>f'(x)</math> doesn't have a vertical asymptote at <math>x=0</math>.

===Horizontal asymptotes===
[[File:Asymptote03.svg|thumb|400px|The [[arctangent]] function has two different asymptotes.]]

''Horizontal asymptotes'' are horizontal lines that the graph of the function approaches as {{math|''x'' &rarr; ±&infin;}}.  The horizontal line ''y''&nbsp;=&nbsp;''c'' is a horizontal asymptote of the function ''y''&nbsp;=&nbsp;''ƒ''(''x'') if
:<math>\lim_{x\rightarrow -\infty}f(x)=c</math> or <math>\lim_{x\rightarrow +\infty}f(x)=c</math>.
In the first case, ''ƒ''(''x'') has ''y''&nbsp;=&nbsp;''c'' as asymptote when ''x'' tends to {{math|&minus;∞}}, and in the second ''ƒ''(''x'') has ''y''&nbsp;=&nbsp;''c'' as an asymptote as ''x'' tends to {{math|+∞}}.

For example, the [[arctangent]] function satisfies
:<math>\lim_{x\rightarrow -\infty}\arctan(x)=-\frac{\pi}{2}</math> and <math>\lim_{x\rightarrow+\infty}\arctan(x)=\frac{\pi}{2}.</math>

So the line {{math|1=''y'' = –{{pi}}/2}} is a horizontal asymptote for the arctangent when ''x'' tends to {{math|–∞}}, and {{math|1=''y'' = {{pi}}/2}} is a horizontal asymptote for the arctangent when ''x'' tends to {{math|+∞}}.

Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function {{math|1=ƒ(''x'') = 1/(''x''<sup>2</sup>+1)}} has a horizontal asymptote at ''y''&nbsp;=&nbsp;0 when ''x'' tends both to {{math|&minus;∞}} and {{math|+∞}} because, respectively,
:<math>\lim_{x\to -\infty}\frac{1}{x^2+1}=\lim_{x\to +\infty}\frac{1}{x^2+1}=0.</math>

Other common functions that have one or two horizontal asymptotes include {{math|''x'' ↦ 1/''x''}} (that has an [[hyperbola]] as it graph), the [[Gaussian function]] <math>x\mapsto \exp(-x^2),</math> the [[error function]], and the [[logistic function]].

===Oblique asymptotes===
[[File:1-over-x-plus-x.svg|right|thumb|220px|In the graph of <math>f(x) = x+\tfrac{1}{x}</math>, the ''y''-axis (''x'' = 0) and the line ''y'' = ''x'' are both asymptotes.]]
When a linear asymptote is not parallel to the ''x''- or ''y''-axis, it is called an ''oblique asymptote'' or ''slant asymptote''. A function ''ƒ''(''x'') is asymptotic to the straight line {{nowrap|1=''y'' = ''mx'' + ''n''}} (''m''&nbsp;≠&nbsp;0) if

:<math>\lim_{x \to +\infty}\left[ f(x)-(mx+n)\right] = 0 \, \mbox{ or } \lim_{x \to -\infty}\left[ f(x)-(mx+n)\right] = 0.</math>

In the first case the line {{nowrap|1=''y'' = ''mx'' + ''n''}} is an oblique asymptote of ''ƒ''(''x'') when ''x'' tends to +∞, and in the second case the line {{nowrap|1=''y'' = ''mx'' + ''n''}} is an oblique asymptote of ''ƒ''(''x'') when ''x'' tends to &minus;∞.

An example is ''ƒ''(''x'')&nbsp;=&nbsp;''x'' + 1/''x'', which has the oblique asymptote ''y''&nbsp;=&nbsp;''x'' (that is ''m''&nbsp;=&nbsp;1, ''n''&nbsp;=&nbsp;0) as seen in the limits
:<math>\lim_{x\to\pm\infty}\left[f(x)-x\right]</math>
:<math>=\lim_{x\to\pm\infty}\left[\left(x+\frac{1}{x}\right)-x\right]</math>
:<math>=\lim_{x\to\pm\infty}\frac{1}{x}=0.</math>