Answer
$y=9/2$
Work Step by Step
Since
\begin{align*}
\lim _{x \rightarrow \infty} \frac{9 x^{2}-4}{2 x^{2}-x}&=\lim _{x \rightarrow \infty} \frac{\frac{9 x^{2}}{x^{2}}-\frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}}-\frac{x}{x^{2}}}\\
&=\lim _{x \rightarrow \infty} \frac{9-\frac{4}{x^{2}}}{2-\frac{1}{x}}\\
&=\frac{9-\frac{4}{\infty}}{2-\frac{1}{\infty}}\\
&=\frac{9}{2}
\end{align*}
and
\begin{align*}
\lim _{x \rightarrow- \infty} \frac{9 x^{2}-4}{2 x^{2}-x}&=\lim _{x \rightarrow -\infty} \frac{\frac{9 x^{2}}{x^{2}}-\frac{4}{x^{2}}}{\frac{2 x^{2}}{x^{2}}-\frac{x}{x^{2}}}\\
&=\lim _{x \rightarrow- \infty} \frac{9-\frac{4}{x^{2}}}{2-\frac{1}{x}}\\
&=\frac{9-\frac{4}{-\infty}}{2-\frac{1}{-\infty}}\\
&=\frac{9}{2}
\end{align*}
Then, the horizontal asymptote is $y=9/2$.
