Answer
See below:
Work Step by Step
Consider the function $ y=\frac{1}{x}$,
For $\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)$,
Approach the point $ x=-1$ on the graph from the left side; the value of the $ y-$ coordinate approaches to $-1$. So, $\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)=-1$,
For $\underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)$,
Now approach the point from the right side. As the point is reached, the value of the $ y-$ coordinate approaches to $-1$. So, $\underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)=-1$,
Since, $\underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)=\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)=-1$, the value for the given limit exists and is given by,
$\underset{x\to -1}{\mathop{\lim }}\,\frac{1}{x}=-1$
