# Combination Theorem for Limits of Functions/Multiple Rule

## 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:

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

$\blacksquare$