Limit is multiplicative

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). In that case, the limit of the pointwise product of functions $f\cdot g$ exists and is the sum of the individual limits:

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

Equivalenty:

$\underset{x\to c}{lim}[f(x)g(x)]=\underset{x\to c}{lim}f(x)\cdot \underset{x\to c}{lim}g(x)$