Answer
The statement "I am working with functions $ f $ and $ g $ for which $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$, $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=0$, and $\underset{x\to 4}{\mathop{\lim }}\,\frac{g\left( x \right)}{f\left( x \right)}\ne 0$ " makes sense.
Work Step by Step
According to the quotient property of limits,
$\underset{x\to a}{\mathop{\lim }}\,\frac{f\left( x \right)}{g\left( x \right)}=\frac{\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)}{\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)},\text{ }if\underset{x\to a}{\mathop{\lim }}\,g\left( x \right)\ne 0$
But here, $\underset{x\to 4}{\mathop{\lim }}\,f\left( x \right)=0$, $\underset{x\to 4}{\mathop{\lim }}\,g\left( x \right)=0$
So, the quotient property of limits cannot be applied.
Thus, $\underset{x\to 4}{\mathop{\lim }}\,\frac{g\left( x \right)}{f\left( x \right)}\ne 0$
Thus, the given statement makes sense.
