Answer
The limit $\lim _{x \rightarrow 4} \frac{1}{(x-4)^{3}}$ does not exist.
Work Step by Step
We have $$
\lim _{x \rightarrow 4} \frac{1}{(x-4)^{3}}= \frac{1}{0}.
$$
Which is undefined. Now, we check the one-sided limits.
$$ \lim _{x \rightarrow 4^+} \frac{1}{(x-4)^{3}}=\infty,\quad \lim _{x \rightarrow 4^-} \frac{1}{(x-4)^{3}}=-\infty.$$
Hence, the limit $\lim _{x \rightarrow 4} \frac{1}{(x-4)^{3}}$ does not exist.
