Combination Theorem for Limits of Functions/Combined Sum 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:

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


Let $\lambda, \mu \in \R$ be arbitrary real numbers.


Then:

$\ds \lim_{x \mathop \to c} \paren {\lambda \map f x + \mu \map g x} = \lambda l + \mu m$


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:

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


Let $\lambda, \mu \in \C$ be arbitrary complex numbers.


Then:

$\ds \lim_{z \mathop \to c} \paren {\lambda \map f z + \mu \map g z} = \lambda l + \mu m$