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