Combination Theorem for Limits of Functions/Multiple Rule

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Theorem

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:

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


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:

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