Combination Theorem for Limits of Functions/Product Rule

Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ and $g$ be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:

$\displaystyle \lim_{x \mathop \to c} \map f x = l$
$\displaystyle \lim_{x \mathop \to c} \map g x = m$

Then:

$\displaystyle \lim_{x \mathop \to c} \ \paren {\map f x \map g x} = l m$

Proof

Let $\sequence {x_n}$ be any sequence of points of $S$ such that:

$\forall n \in \N: x_n \ne c$
$\displaystyle \lim_{n \mathop \to \infty} x_n = c$
$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = l$
$\displaystyle \lim_{n \mathop \to \infty} \map g {x_n} = m$

By the Product Rule for Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \ \paren {\map f {x_n} \map g {x_n} } = l m$

Applying Limit of Function by Convergent Sequences again, we get:

$\displaystyle \lim_{x \mathop \to c} \paren {\map f x \map g x} = l m$

$\blacksquare$