Answer
The value of $\underset{x\to 1}{\mathop{\lim }}\,\left[ f\left( x \right)+g\left( x \right) \right]$ is $-2$.
Work Step by Step
To find the value of $\underset{x\to 1}{\mathop{\lim }}\,\left[ f\left( x \right)+g\left( x \right) \right]$, find the value of $\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)$ and the value of $\underset{x\to 1}{\mathop{\lim }}\,g\left( x \right)$.
It is seen from the graph that as the value of x nears $1$ from the left or right, the value of the function $ f\left( x \right)$ nears $-1$.
Thus, $\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=-1$.
It is seen from the graph that as the value of x nears $1$ from the left or right, the value of the function $ g\left( x \right)$ nears $-1$.
Thus, $\underset{x\to 1}{\mathop{\lim }}\,g\left( x \right)=-1$.
Now find the value of $\underset{x\to 1}{\mathop{\lim }}\,\left[ f\left( x \right)+g\left( x \right) \right]$,
$\begin{align}
& \underset{x\to 1}{\mathop{\lim }}\,\left[ f\left( x \right)+g\left( x \right) \right]=\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to 1}{\mathop{\lim }}\,g\left( x \right) \\
& =-1+\left( -1 \right) \\
& =-2
\end{align}$
Thus, the value of $\underset{x\to 1}{\mathop{\lim }}\,\left[ f\left( x \right)+g\left( x \right) \right]$ is $-2$.
