Sample Problem

Determine the vertical and horizontal asymptotes of f(x)=8x2/(x2-1).

Horizontal asymptote at y=

Vertical asymptotes at x= and x= (order from left to right)


The graph of f has one horizontal asymptote because the degrees of the leading

variables of the numerator and denominator are the same (both have x2 as biggest degree);

Since the degrees are the same, the graph of f has the line y=8/1 (the ratio of the leading coefficients) as a horizontal asymptote.

y=8 : horizontal asymptote

The graph of f has vertical asymptotes at the zeros of the denominator (D(x)).

The zeros of D(x)=x2-1=(x+1)(x-1) are at x=-1 and x=1.

x=-1 and x=1 : vertical asymptotes