Editing Asymptote (section)

===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 +∞.