Combination Theorem for Limits of Functions

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:

$\displaystyle \lim_{x \to c} \ f \left({x}\right) = l$
$\displaystyle \lim_{x \to c} \ g \left({x}\right) = m$

Let $\lambda, \mu \in X$ be arbitrary numbers in $X$.

Then the following results hold:

Sum Rule

$\displaystyle \lim_{x \to c} \ \left({f \left({x}\right) + g \left({x}\right)}\right) = l + m$

Multiple Rule

$\displaystyle \lim_{x \to c} \left({\lambda f \left({x}\right)}\right) = \lambda l$

Combined Sum Rule

$\displaystyle \lim_{x \to c} \left({\lambda f \left({x}\right) + \mu g \left({x}\right)}\right) = \lambda l + \mu m$

Product Rule

$\displaystyle \lim_{x \to c} \ \left({f \left({x}\right) g \left({x}\right)}\right) = l m$

Quotient Rule

$\displaystyle \lim_{x \to c} \frac {f \left({x}\right)} {g \left({x}\right)} = \frac l m$

provided that $m \ne 0$.

(In the case that $l = m = 0$, see L'Hôpital's Rule).