Limit of quotient equals quotient of limits

Statement

Suppose $f$ and $g$ are functions of one variable. Suppose $c\in \mathbb{R}$ is such that both $f$ and $g$ are defined on the immediate left and the immediate right of $c$. Further, suppose that the limits $\underset{x\to c}{lim}f(x)$ and $\underset{x\to c}{lim}g(x)$ both exist (as finite numbers) and that $\underset{x\to c}{lim}g(x)\ne 0$. In that case, the limit of the pointwise quotient of functions $f/g$ exists at $c$ and is the quotient of the individual limits:

$\underset{x\to c}{lim}(\frac{f}{g})(x)=\frac{\underset{x\to c}{lim}f(x)}{\underset{x\to c}{lim}g(x)}$

Equivalenty:

$\underset{x\to c}{lim}\frac{f(x)}{g(x)}=\frac{\underset{x\to c}{lim}f(x)}{\underset{x\to c}{lim}g(x)}$