Combination Theorem for Limits of Functions/Multiple Rule

Real Functions
Let $\R$ denote the real numbers.

Let $f$ be a real function defined on an open subset $S \subseteq \R$, except possibly at the point $c \in S$.

Let $f$ tend to the following limit:


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

Let $\lambda \in \R$ be an arbitrary real number.

Then:

Complex Functions
Let $\C$ denote the complex numbers.

Let $f$ be a complex function defined on an open subset $S \subseteq \C$, except possibly at the point $c \in S$.

Let $f$ tend to the following limit:


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

Let $\lambda \in \C$ be an arbitrary complex number.

Then: