Combination Theorem for Limits of Functions/Real/Quotient Rule

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:

$\ds \lim_{x \mathop \to c} \map f x = l$

$\ds \lim_{x \mathop \to c} \map g x = m$

Then:

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

provided that $m \ne 0$.

Proof

Let $\sequence {x_n}$ be any sequence of elements of $S$ such that:

$\forall n \in \N_{>0}: x_n \ne c$

$\ds \lim_{n \mathop \to \infty} x_n = c$

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 Quotient Rule for Real Sequences:

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

provided that $m \ne 0$.

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

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

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 $1 \ \text{(iii)}$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.4$: Continuity
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 8.12 \ \text{(iii)}$