Editing Asymptote (section)

===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]].