Answer
The value of $\underset{x\to -1}{\mathop{\lim }}\,f\left( x \right)$ does not exist.
Work Step by Step
The value of $\underset{x\to -1}{\mathop{\lim }}\,f\left( x \right)$ exists only if the values of $\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ and $\underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ exist and are equal.
It is seen from the graph that, the value of $\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)$ is $2$ and the value of $\underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)$ is $1$.
Since $\underset{x\to -{{1}^{-}}}{\mathop{\lim }}\,f\left( x \right)\ne \underset{x\to -{{1}^{+}}}{\mathop{\lim }}\,f\left( x \right)$,
Thus, the value of $\underset{x\to -1}{\mathop{\lim }}\,f\left( x \right)$ does not exist.
