Editing Asymptote (section)

==== 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.