Answer
The left and right limits are equal when $n$ is even.
Work Step by Step
Consider the case when $n$ is even (e.g. $1/x^2$),
\begin{align*}
\lim _{x \rightarrow 0-} \frac{1}{x^{n}}&=\infty\\
\lim _{x \rightarrow 0+} \frac{1}{x^{n}}&=\infty
\end{align*}
and for $n$ is odd (e.g. $1/x^3$),
\begin{align*}
\lim _{x \rightarrow 0-} \frac{1}{x^{n}}&=-\infty\\
\lim _{x \rightarrow 0+} \frac{1}{x^{n}}&=\infty
\end{align*}
We see that the left and right limits are equal for even powers of $n$, but not for odd powers.
