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 $\lim _{x\to c}f(x)$ and $\lim _{x\to c}g(x)$ both exist (as finite numbers) and that $\lim _{x\to c}g(x)\neq 0$. In that case, the limit of the pointwise quotient of functions $f/g$ exists at $c$ and is the quotient of the individual limits:

$\lim _{x\to c}\left({\frac {f}{g}}\right)(x)={\frac {\lim _{x\to c}f(x)}{\lim _{x\to c}g(x)}}$

Equivalenty:

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