Combination Theorem for Limits of Functions/Quotient Rule

Theorem

Real Functions

Let $\R$ denote the real numbers.

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

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

$\ds \lim_{x \mathop \to c} \map f x = l$

$\ds \lim_{x \mathop \to c} \map g x = m$

Then:

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

provided that $m \ne 0$.

Complex Functions

Let $\C$ denote the complex numbers.

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

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

$\ds \lim_{z \mathop \to c} \map f z = l$

$\ds \lim_{z \mathop \to c} \map g z = m$

Then:

$\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$

provided that $m \ne 0$.