# Combination Theorem for Limits of Functions/Sum Rule

Jump to navigation
Jump to search

## 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

- 1942: James M. Hyslop:
*Infinite Series*: $\S 4$: Theorem $1 \ \text{(i)}$