Answer
The value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$ is $-1$.
Work Step by Step
To find the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$, find the value of $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)$ and the value of $\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right)$
It is seen from the graph that as the value of x nears $0$ from the left or right, the value of the function $ f\left( x \right)$ nears $0$.
Thus, $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=0$
It is seen from the graph that as the value of x nears $0$ from the left or right, the value of the function $ g\left( x \right)$ nears $1$.
Thus, $\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right)=1$
Now find the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$,
$\begin{align}
& \underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]=\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)-\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right) \\
& =0-1 \\
& =-1
\end{align}$
Thus, the value of $\underset{x\to 0}{\mathop{\lim }}\,\left[ f\left( x \right)-g\left( x \right) \right]$ is $-1$.
