Combination Theorem for Limits of Functions/Quotient 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:

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

provided that $m \ne 0$.

(In the case that $l = m = 0$, see L'Hôpital's Rule).

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:

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

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

By the Quotient Rule for Sequences:

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

provided that $m \ne 0$.

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

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

provided that $m \ne 0$.

$\blacksquare$

Sources

• James M. Hyslop: Infinite Series (1942): $\S 4$: Theorem $1 \ \text{(iii)}$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.12 \ \text{(iii)}$