Combination Theorem for Limits of Functions/Complex/Quotient Rule

Theorem

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 $m \ne 0$.

Proof

Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:

$\forall n \in \N_{>0}: z_n \ne c$

$\ds \lim_{n \mathop \to \infty} z_n = c$

By Limit of Real Function by Convergent Sequences:

$\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$

$\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$

By the Quotient Rule for Real Sequences:

$\ds \lim_{n \mathop \to \infty} \frac {\map f {z_n} } {\map g {z_n} } = \frac l m$

provided that $m \ne 0$.

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

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

provided that $m \ne 0$.

$\blacksquare$