Combination Theorem for Limits of Functions/Complex
Jump to navigation
Jump to search
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:
- $\ds \lim_{z \mathop \to c} \map f z = l$
- $\ds \lim_{z \mathop \to c} \map g z = m$
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then the following results hold:
Sum Rule
- $\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$
Multiple Rule
- $\ds \lim_{z \mathop \to c} \lambda \map f z = \lambda l$
Combined Sum Rule
- $\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$
Product Rule
- $\ds \lim_{z \mathop \to c} \ \paren {\map f z \map g z} = l m$
Quotient Rule
- $\ds \lim_{z \mathop \to c} \frac {\map f z} {\map g z} = \frac l m$
provided that $m \ne 0$.