Answer
The value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{4g\left( x \right)}{{{\left[ f\left( x \right) \right]}^{2}}}$ is $4$.
Work Step by Step
To find the value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{4g\left( x \right)}{{{\left[ f\left( x \right) \right]}^{2}}}$, 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 table that as the value of x nears $0$ from the left or right, the value of the function $ f\left( x \right)$ nears $-1$.
Thus, $\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=-1$
It is seen from the table 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 $\underset{x\to 0}{\mathop{\lim }}\,\frac{4g\left( x \right)}{{{\left[ f\left( x \right) \right]}^{2}}}$,
$\begin{align}
& \underset{x\to 0}{\mathop{\lim }}\,\frac{4g\left( x \right)}{{{\left[ f\left( x \right) \right]}^{2}}}=\frac{4\underset{x\to 0}{\mathop{\lim }}\,g\left( x \right)}{{{\left[ \underset{x\to 0}{\mathop{\lim }}\,f\left( x \right) \right]}^{2}}} \\
& =\frac{4\left( 1 \right)}{{{\left[ -1 \right]}^{2}}} \\
& =\frac{4}{1} \\
& =4
\end{align}$
Thus, the value of $\underset{x\to 0}{\mathop{\lim }}\,\frac{4g\left( x \right)}{{{\left[ f\left( x \right) \right]}^{2}}}$ is $4$.
