Answer
The statement "I am working with functions $ f $ and $ g $ for which $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$, $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=-5$, and $\underset{x\to 4}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=-5$ " makes sense.
Work Step by Step
According to the limit property,
$\underset{x\to a}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)$.
So, here $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$ and $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=-5$.
Thus,
$\begin{align}
& \underset{x\to 4}{\mathop{\lim }}\,\left( f\left( x \right)+g\left( x \right) \right)=\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)+\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right) \\
& =0+\left( -5 \right) \\
& =-5
\end{align}$.
Thus, the given statement makes sense.
