Calculus (3rd Edition)

The limit $\lim _{x \rightarrow 4} \frac{1}{(x-4)^{3}}$ does not exist.
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.