Find the Asymptotes y=(3x^2+1)/(x^2-5)

Find where the expression ${\displaystyle \frac{3x^2+1}{x^2-5}}$ is undefined.

Since ${\displaystyle \frac{3x^2+1}{x^2-5}}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -\sqrt{5}}$ from the left and ${\displaystyle \frac{3x^2+1}{x^2-5}}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle -\sqrt{5}}$ from the right, then ${\displaystyle x=-\sqrt{5}}$ is a vertical asymptote.

Since ${\displaystyle \frac{3x^2+1}{x^2-5}}$${\displaystyle \to }$${\displaystyle -\infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle \sqrt{5}}$ from the left and ${\displaystyle \frac{3x^2+1}{x^2-5}}$${\displaystyle \to }$${\displaystyle \infty }$ as ${\displaystyle x}$${\displaystyle \to }$${\displaystyle \sqrt{5}}$ from the right, then ${\displaystyle x=\sqrt{5}}$ is a vertical asymptote.

List all of the vertical asymptotes:

Consider the rational function ${\displaystyle R\left(x\right)=\frac{a{x}^n}{b{x}^m}}$ where ${\displaystyle n}$ is the degree of the numerator and ${\displaystyle m}$ is the degree of the denominator.

Find ${\displaystyle n}$ and ${\displaystyle m}$.

Since ${\displaystyle n=m}$, the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$ where ${\displaystyle a=3}$ and ${\displaystyle b=1}$.

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

This is the set of all asymptotes.