Prove that \begin{equation}\lim_{x\to a}\,\frac{1}{f(x)}=\frac{1}{L}.\end{equation}

Problem: 

Prove that \begin{equation}\lim_{x\to a}\,\frac{1}{f(x)}=\frac{1}{L}\end{equation} if
\begin{equation}
\lim_{x\to a}\,f(x)=L
\end{equation}
and $L\neq 0$.

Answer: 

It is true that \begin{equation}\lim_{x\to a}\,\frac{1}{f(x)}=\frac{1}{L}.\end{equation}