Question

How do you find a vertical asymptote for a rational function?

Answer 1

Let `f(x)=(p(x))/(q(x))` be a rational function. A line `x=x_0` is a vertical asymptote of `f` when

`lim_(x->x_0^+-)(p(x))/(q(x))=+-infty.`

Since a rational function is continuous in its domain, the possible vertical asymptote `x=x_0` are among that for which `q(x_0)=0`.

In other words, first we have to find a point `x_0` that is not in the domain of `f`, ie, `q(x_0)=0`, and then verify if limits of `f` are `+-infty` when x goes to `x_0^+` and `x_0^-`.