Rational Functions with Horizontal Asymptotes

Rational function of the form \(y=\dfrac{ax+b}{cx+d}\)

Long division converts \(y=\dfrac{ax+b}{cx+d}\), where \(x\ne-\dfrac{d}{c}\) and \(c\ne0\), into \(y=\dfrac{a}{c}+\dfrac{k}{cx+d}\), with \(k=\dfrac{bc-ad}{c}\). Combining the terms reverses this conversion.

Horizontal asymptote: \(y=\dfrac{a}{c}\), since \(x\to\pm\infty\Rightarrow\dfrac{k}{cx+d}\to0\).

If \(ad-bc\ne0\), the vertical asymptote is \(x=-\dfrac{d}{c}\). If \(ad-bc=0\), the expression equals \(a/c\) for \(x\ne-d/c\); the excluded point is a hole, not a vertical asymptote.