Combination Theorem for Limits of Functions/Combined Sum Rule

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:

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: