Combination Theorem for Limits of Functions/Sum Rule

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Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $f$ and $g$ be functions defined on an open subset $S \subseteq X$, except possibly at the point $c \in S$.

Let $f$ and $g$ tend to the following limits:

$\displaystyle \lim_{x \mathop \to c} \map f x = l$
$\displaystyle \lim_{x \mathop \to c} \map g x = m$


Then:

$\displaystyle \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$


Proof

Let $\sequence {x_n}$ be any sequence of points of $S$ such that:

$\forall n \in \N^*: x_n \ne c$
$\displaystyle \lim_{n \mathop \to \infty} \ x_n = c$


By Limit of Function by Convergent Sequences, we have:

$\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = l$
$\displaystyle \lim_{n \mathop \to \infty} \map g {x_n} = m$


By the Sum Rule for Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {\map f {x_n} + \map g {x_n} } = l + m$


Applying Limit of Function by Convergent Sequences again, we get:

$\displaystyle \lim_{x \mathop \to c} \paren {\map f x + \map g x} = l + m$

$\blacksquare$


Sources