# 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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- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 8.12 \ \text{(i)}$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): Appendix: $\S 18.6$: Limits of functions