Combination Theorem for Limits of Functions/Complex/Sum Rule
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Theorem
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:
- $\ds \lim_{z \mathop \to c} \map f z = l$
- $\ds \lim_{z \mathop \to c} \map g z = m$
Then:
- $\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$
Proof
Let $\sequence {z_n}$ be any sequence of elements of $S$ such that:
- $\forall n \in \N_{>0}: z_n \ne c$
- $\ds \lim_{n \mathop \to \infty} z_n = c$
By Limit of Complex Function by Convergent Sequences:
- $\ds \lim_{n \mathop \to \infty} \map f {z_n} = l$
- $\ds \lim_{n \mathop \to \infty} \map g {z_n} = m$
By the Sum Rule for Complex Sequences:
- $\ds \lim_{n \mathop \to \infty} \paren {\map f {z_n} + \map g {z_n} } = l + m$
Applying Limit of Complex Function by Convergent Sequences again, we get:
- $\ds \lim_{z \mathop \to c} \paren {\map f z + \map g z} = l + m$
$\blacksquare$