Answer
$$
\lim _{x \rightarrow 3^{-}}\left(\frac{1}{x-3}\right) \ \text{does not exist }
$$
Work Step by Step
From the following table
\begin{array}{|c|c|c|c|c|c|}\hline x\to3^- & {2.9} & {2.99} & {2.999} & {2.9999} \\ \hline \frac{1}{x-3} & {-10} & {-100} & {-1000} & {-10000} \\ \hline\end{array}
This means that $$
\lim _{x \rightarrow 3^{-}}\left(\frac{1}{x-3}\right)=-\infty
$$ and
\begin{array}{|c|c|c|c|c|}\hline x & {3.1} & {3.01} & {3.001} & {3.0001} \\ \hline 1 / x-3 & {10} & {100} & {1000} & {10000} \\ \hline\end{array}
This means that $$
\lim _{x \rightarrow 3^{+}}\left(\frac{1}{x-3}\right)=\infty
$$
Hence: $$
\lim _{x \rightarrow 3^{-}}\left(\frac{1}{x-3}\right) \ \text{does not exists }
$$
