# Combination Theorem for Limits of Functions/Quotient Rule

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## Contents

## 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:

- $\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$

provided that $m \ne 0$.

## Proof

Let $\sequence {x_n}$ 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:

- $\displaystyle \lim_{n \mathop \to \infty} \map f {x_n} = l$
- $\displaystyle \lim_{n \mathop \to \infty} \map g {x_n} = m$

By the Quotient Rule for Sequences:

- $\displaystyle \lim_{n \mathop \to \infty} \frac {\map f {x_n} } {\map g {x_n} } = \frac l m$

provided that $m \ne 0$.

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

- $\displaystyle \lim_{x \mathop \to c} \frac {\map f x} {\map g x} = \frac l m$

provided that $m \ne 0$.

$\blacksquare$

## Also see

- L'Hôpital's Rule: for the case where $l = m = 0$

## Sources

- 1947: James M. Hyslop:
*Infinite Series*(3rd ed.) ... (previous) ... (next): Chapter $\text I$: Functions and Limits: $\S 4$: Limits of Functions: Theorem $1 \ \text{(iii)}$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 8.12 \ \text{(iii)}$