Answer
$$3$$
Work Step by Step
\begin{align*}
\lim _{x \rightarrow-\infty} \frac{4-3 x^{3}}{\sqrt{x^{6}+9}}&=\lim _{x \rightarrow-\infty} \frac{\left(4-3 x^{3}\right) / \sqrt{x^{6}}}{\sqrt{x^{6}+9} / \sqrt{x^{6}}}\\
&=\lim _{x \rightarrow-\infty} \frac{\left(4-3 x^{3}\right) /\left(-x^{3}\right)}{\sqrt{\left(x^{6}+9\right) / x^{6}}}\\
&=\lim _{x \rightarrow \infty} \frac{\left(-4 / x^{3}+3\right)}{\sqrt{1+9 / x^{6}}}\\
&=\frac{(0+3)}{\sqrt{1+0}}\\
&=3
\end{align*}