Answer
$ f(x)$ does not have any horizontal asymptotes.
Work Step by Step
Since
\begin{align*}
\lim _{x \rightarrow \infty} \frac{x^{2}-3 x^{4}}{x-1}&=\lim _{x \rightarrow \infty} \frac{1 / x^{2}-3}{1 / x^{3}-1 / x^{4}}\\
&=-\infty
\end{align*}
and
\begin{align*}
\lim _{x \rightarrow-\infty} \frac{x^{2}-3 x^{4}}{x-1}&=\lim _{x \rightarrow-\infty} \frac{1 / x^{2}-3}{1 / x^{3}-1 / x^{4}}\\
&=\infty
\end{align*}
We see that both limits are infinite and do not match, so the function $ f(x)$ does not have any horizontal asymptotes.
