Combination Theorem for Limits of Functions/Multiple 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 $f$ tend to the following limit:

$\displaystyle \lim_{x \mathop \to c} f \paren x = l$


Let $\lambda \in X$ be an arbitrary number in $X$.


Then:

$\displaystyle \lim_{x \mathop \to c} \ \paren {\lambda f \paren x} = \lambda l$


Proof

Let $\left \langle {x_n} \right \rangle$ be any sequence of points of $S$ such that:

$\forall n \in \N_{>0}: x_n \ne c$
$\displaystyle \lim_{n \mathop \to \infty} x_n = c$


By Limit of Function by Convergent Sequences:

$\displaystyle \lim_{n \mathop \to \infty} f \paren {x_n} = l$


By the Multiple Rule for Sequences:

$\displaystyle \lim_{n \mathop \to \infty} \paren {\lambda f \paren {x_n} } = \lambda l$


Applying Limit of Function by Convergent Sequences again:

$\displaystyle \lim_{x \mathop \to c} \paren {\lambda f \paren x} = \lambda l$

$\blacksquare$