Combination Theorem for Limits of Functions/Real/Combined Sum Rule

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Theorem

Let $\R$ denote the real numbers.

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

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

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


Let $\lambda, \mu \in \R$ be arbitrary real numbers.


Then:

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


Proof

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

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


By Limit of Real Function by Convergent Sequences:

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


By the Combined Sum Rule for Real Sequences:

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


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

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

$\blacksquare$


Sources


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