Question

How do you find the slant asymptote of `(x^3-1)/(x^2-9)`?

Answer 1

It is obvious that as factors of denominator are `(x+3)(x-3)`, the function as two vertical asymptotes `x=3` and `x=-3`.

Horizontal asymptotes are there if highest degrees of numerator and denominator are same. For example, if highest term in numerator is `ax^n` and in denominator is `bx^n`, vertical asymptote will be `y=a/b`. But this is not so here.

Here the degree of numerator is just one more than that of denominator and ratio of highest degree of numerator and denominator is `x3/x^2=x`.

Hence, we have a slanting asymptote `y=x`.

graph{(x^3-1)/(x^2-9) [-32.47, 32.47, -16.24, 16.24]}