Find the Asymptotes f(x)=(x^2-9)/(2x^2+1)

Find where the expression ${\displaystyle \frac{x^2-9}{2x^2+1}}$ is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

The vertical asymptotes occur at areas of infinite discontinuity.

No 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.

1. If ${\displaystyle n<m}$, then the x-axis, ${\displaystyle y=0}$, is the horizontal asymptote.

2. If ${\displaystyle n=m}$, then the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$.

3. If ${\displaystyle n>m}$, then there is no horizontal asymptote (there is an oblique asymptote).

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

Since ${\displaystyle n=m}$, the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$ where ${\displaystyle a=1}$ and ${\displaystyle b=2}$.

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

No Oblique Asymptotes

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: ${\displaystyle y=\frac{1}{2}}$

No Oblique Asymptotes