pagtuboy2b

2023-03-03

How many vertical asymptotes can a rational function have?

helixl3u

Beginner2023-03-04Added 6 answers

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 $p\left(x\right)$.

Therefore, the number of zeros is the largest number of vertical asymptotes that can exist of $q\left(x\right)$, which is at most the degree of $q\left(x\right)$.

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 $p\left(x\right)$.

Therefore, the number of zeros is the largest number of vertical asymptotes that can exist of $q\left(x\right)$, which is at most the degree of $q\left(x\right)$.