Combination Theorem for Limits of Functions/Quotient Rule

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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 $m \ne 0$.

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


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 $m \ne 0$.

$\blacksquare$


Sources