Answer
The value of $\underset{x\to 1}{\mathop{\lim }}\,\sqrt{10+f\left( x \right)}$ is $3$.
Work Step by Step
To find the value of $\underset{x\to 1}{\mathop{\lim }}\,\sqrt{10+f\left( x \right)}$, find the value of $\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)$.
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, $\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=-1$
Now find the value of $\underset{x\to 1}{\mathop{\lim }}\,\sqrt{10+f\left( x \right)}$,
$\begin{align}
& \underset{x\to 1}{\mathop{\lim }}\,\sqrt{10+f\left( x \right)}=\sqrt{10+\underset{x\to 1}{\mathop{\lim f\left( x \right)}}\,} \\
& =\sqrt{10+\left( -1 \right)} \\
& =\sqrt{9} \\
& =3
\end{align}$
Thus, the value of $\underset{x\to 1}{\mathop{\lim }}\,\sqrt{10+f\left( x \right)}$ is $3$.
