Editing Asymptote (section)

== Elementary methods for identifying asymptotes ==
The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).

===General computation of oblique asymptotes for functions===

The oblique asymptote, for the function ''f''(''x''), will be given by the equation ''y'' = ''mx'' + ''n''. The value for ''m'' is computed first and is given by

:<math>m\;\stackrel{\text{def}}{=}\,\lim_{x\rightarrow a}f(x)/x</math>

where ''a'' is either <math>-\infty</math> or <math>+\infty</math> depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.

Having ''m'' then the value for ''n'' can be computed by

:<math>n\;\stackrel{\text{def}}{=}\,\lim_{x\rightarrow a}(f(x)-mx)</math>

where ''a'' should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining ''m'' exist. Otherwise {{nowrap|1=''y'' = ''mx'' + ''n''}} is the oblique asymptote of ''ƒ''(''x'') as ''x'' tends to ''a''.

For example, the function {{nowrap|1=''&fnof;''(''x'') = (2''x''<sup>2</sup> + 3''x'' + 1)/''x''}} has

:<math>m=\lim_{x\rightarrow+\infty}f(x)/x=\lim_{x\rightarrow+\infty}\frac{2x^2+3x+1}{x^2}=2</math> and then
:<math>n=\lim_{x\rightarrow+\infty}(f(x)-mx)=\lim_{x\rightarrow+\infty}\left(\frac{2x^2+3x+1}{x}-2x\right)=3</math>

so that {{nowrap|1=''y'' = 2''x'' + 3}} is the asymptote of ''ƒ''(''x'') when ''x'' tends to +∞.

The function {{nowrap|1=''&fnof;''(''x'') = ln&thinsp;''x''}} has

:<math>m=\lim_{x\rightarrow+\infty}f(x)/x=\lim_{x\rightarrow+\infty}\frac{\ln x}{x}=0</math> and then

:<math>n=\lim_{x\rightarrow+\infty}(f(x)-mx)=\lim_{x\rightarrow+\infty}\ln x</math>, which does not exist.

So {{nowrap|1=''y'' = ln&thinsp;''x''}} does not have an asymptote when ''x'' tends to +∞.

=== Asymptotes for rational functions ===
A [[rational function]] has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.

The [[Degree of a polynomial|degree]] of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.

{| align=center class="wikitable"
|+ The cases of horizontal and oblique asymptotes for rational functions
|-
! deg(numerator)−deg(denominator)
! Asymptotes in general
! Example
! Asymptote for example
|-
| < 0
| <math>y= 0</math>
| <math>f(x)=\frac{1}{x^2+1}</math>
| <math>y=0</math>
|-
| = 0
| ''y'' = the ratio of leading coefficients
| <math>f(x)=\frac{2x^2+7}{3x^2+x+12}</math>
| <math>y=\frac{2}{3}</math>
|-
| = 1
| ''y'' = the quotient of the [[Euclidean division of polynomials|Euclidean division]] of the numerator by the denominator
| <math>f(x)=\frac{2x^2+3x+5}{x}=2x+3+\frac{5}{x}</math>
| <math>y=2x+3</math>
|-
| > 1
| none
| <math>f(x)=\frac{2x^4}{3x^2+1}</math>
| no linear asymptote, but a [[#Curvilinear_asymptotes|curvilinear asymptote]] exists
|}

The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at ''x'' = 0, and ''x'' = 1, but not at ''x'' = 2.
:<math>f(x)=\frac{x^2-5x+6}{x^3-3x^2+2x}=\frac{(x-2)(x-3)}{x(x-1)(x-2)}</math>

==== Oblique asymptotes of rational functions ====
[[File:SlantAsymptoteError.svg|right|thumb|320px|Black: the graph of <math>f(x)=(x^2+x+1)/(x+1)</math>. Red: the asymptote <math>y=x</math>. Green: difference between the graph and its asymptote for <math>x=1,2,3,4,5,6</math>.]]

When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after [[Polynomial long division|dividing]] the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function
:<math>f(x)=\frac{x^2+x+1}{x+1}=x+\frac{1}{x+1}</math>
shown to the right. As the value of ''x'' increases, ''f'' approaches the asymptote ''y'' = ''x''. This is because the other term,  1/(''x''+1), approaches 0.

If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as ''x'' increases, but the quotient will not be linear, and the function does not have an oblique asymptote.

=== Transformations of known functions ===
If a known function has an asymptote (such as ''y''=0 for ''f''(x)=''e''<sup>''x''</sup>), then the translations of it also have an asymptote.
* If ''x''=''a'' is a vertical asymptote of ''f''(''x''), then ''x''=''a''+''h'' is a vertical asymptote of ''f''(''x''-''h'')
* If ''y''=''c'' is a horizontal asymptote of ''f''(''x''), then ''y''=''c''+''k'' is a horizontal asymptote of ''f''(''x'')+''k''

If a known function has an asymptote, then the [[Homothetic transformation|scaling]] of the function also have an asymptote.

* If ''y''=''ax''+''b'' is an asymptote of ''f''(''x''), then ''y''=''cax''+''cb'' is an asymptote of ''cf''(''x'')
For example, ''f''(''x'')=''e''<sup>''x''-1</sup>+2 has horizontal asymptote ''y''=0+2=2, and no vertical or oblique asymptotes.