Question

How many vertical asymptotes can a rational function have?

Answer 1

Assuming that you mean a function of the form `(p(x))/(q(x))` where `p(x)` and `q(x)` are polynomials, then the maximum number of vertical asymptotes is the degree of `q(x)`.

Explanation:

Suppose `f(x) = (p(x))/(q(x))` where `p(x)` and `q(x)` are polynomials.

If `q(x_1) = 0` and `p(x_1) != 0` then `f(x)` will have a vertical asymptote at `x = x_1`.

If `q(x_1) = 0` and `p(x_1) = 0` then `f(x)` will have a vertical asymptote at `x = x_1` if and only if the multiplicity of this zero of `q(x)` is greater than the multiplicity of this zero of `p(x)`.

So the maximum possible number of vertical asymptotes is the number of zeros of `q(x)`, which is at most the degree of `q(x)`.