# How many vertical asymptotes can a rational function have?

Aug 20, 2015

Assuming that you mean a function of the form $\frac{p \left(x\right)}{q \left(x\right)}$ where $p \left(x\right)$ and $q \left(x\right)$ are polynomials, then the maximum number of vertical asymptotes is the degree of $q \left(x\right)$.

#### Explanation:

Suppose $f \left(x\right) = \frac{p \left(x\right)}{q \left(x\right)}$ where $p \left(x\right)$ and $q \left(x\right)$ are polynomials.

If $q \left({x}_{1}\right) = 0$ and $p \left({x}_{1}\right) \ne 0$ then $f \left(x\right)$ will have a vertical asymptote at $x = {x}_{1}$.

If $q \left({x}_{1}\right) = 0$ and $p \left({x}_{1}\right) = 0$ then $f \left(x\right)$ will have a vertical asymptote at $x = {x}_{1}$ if and only if the multiplicity of this zero of $q \left(x\right)$ is greater than the multiplicity of this zero of $p \left(x\right)$.

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