Answer
\begin{align*}
\lim _{x \rightarrow -1^{-}} f(x)&=-\infty \\
\lim _{x \rightarrow -1^{+}} f(x)&=\infty \\
\lim _{x \rightarrow -1} f(x)&= does~not~exist \\
\lim _{x \rightarrow 3^{-}} f(x)&= +\infty\\
\lim _{x \rightarrow 3^{+}} f(x)&= +\infty\\
\lim _{x \rightarrow 3^{}} f(x)&= +\infty\\
\lim _{x \rightarrow 5^{-}} f(x)&= -\infty\\
\lim _{x \rightarrow 5^{+}} f(x)&= -\infty\\
\lim _{x \rightarrow 5^{}} f(x)&=-\infty\\
\end{align*}
Work Step by Step
We find the limits by observing the figure. For the overall limit to exist, the left and right limits must equal each other. Thus, we have:
\begin{align*}
\lim _{x \rightarrow -1^{-}} f(x)&=-\infty \\
\lim _{x \rightarrow -1^{+}} f(x)&=\infty \\
\lim _{x \rightarrow -1} f(x)&= does~not~exist \\
\lim _{x \rightarrow 3^{-}} f(x)&= +\infty\\
\lim _{x \rightarrow 3^{+}} f(x)&= +\infty\\
\lim _{x \rightarrow 3^{}} f(x)&= +\infty\\
\lim _{x \rightarrow 5^{-}} f(x)&= -\infty\\
\lim _{x \rightarrow 5^{+}} f(x)&= -\infty\\
\lim _{x \rightarrow 5^{}} f(x)&=-\infty\\
\end{align*}
