Combination Theorem for Limits of Functions/Quotient Rule
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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 $m \ne 0$.