# Combination Theorem for Limits of Functions/Sum 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 $f$ and $g$ tend to the following limits:

- $\displaystyle \lim_{x \to c} \ f \left({x}\right) = l$
- $\displaystyle \lim_{x \to c} \ g \left({x}\right) = m$

Then:

- $\displaystyle \lim_{x \to c} \ \left({f \left({x}\right) + g \left({x}\right)}\right) = l + m$

## Proof

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

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

By Limit of Function by Convergent Sequences, we have:

- $\displaystyle \lim_{n \to \infty} \ f \left({x_n}\right) = l$
- $\displaystyle \lim_{n \to \infty} \ g \left({x_n}\right) = m$

By the Sum Rule for Sequences:

- $\displaystyle \lim_{n \to \infty} \ \left({f \left({x_n}\right) + g \left({x_n}\right)}\right) = l + m$

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

- $\displaystyle \lim_{x \to c} \ \left({f \left({x}\right) + g \left({x}\right)}\right) = l + m$

$\blacksquare$

## Sources

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