In search of asymptotes

Dividing polynomials ---------------------

A classic problem for anyone studying functions is to find the non-vertical asymptotes of a rational function A / B, where A and B are polynomials of degree d₁ and d₂ respectively. Euclidean division of A by B yields a quotient Q (of degree d₁ – d₂) and a remainder R (of degree strictly less than d₂) such that A = BQ + R. We thus have:

In the cases where the difference d₁ – d₂ equals 0 or 1, the quotient Q can be written in the form mx + p; moreover, the term R / B then tends to 0 as the variable x tends to infinity (both toward + 3 and toward – 3). Thus, when the variable x becomes very large in absolute value, A / B behaves approximately like the function that maps x to mx + p. The line with equation y = mx + p is an asymptote to the rational function A / B. This line is horizontal when m = 0 (that is, when d₁ = d₂); it is oblique when m is nonzero (which occurs when d₁ = d₂ + 1).

Let's find the asymptotes of the function f defined by (2x³ – 3x² + x – 1) / (x² + 1). There is no vertical asymptote, since the denominator never vanishes, nor any horizontal asymptote, since the degrees of the numerator and denominator do not coincide. However, there is indeed an oblique asymptote: f (x) = 2x – 3 + (2 – x) / (x² + 1). The line with equation y = 2x – 3 is an asymptote.