A proof of this lemma proceeds by applying the Cauchy mean value theorem to a relevant ratio, noting that two of the difference terms vanish, then shrinking the interval. A hypothesis in this theorem is that the relevant functions oscillate at most finitely often near the relevant point; this is because a derivative of the 
A proof of this lemma proceeds by applying the Cauchy mean value theorem to a relevant ratio, noting that two of the difference terms vanish, then shrinking the interval. A hypothesis in this theorem is that the relevant functions oscillate at most finitely often near the relevant point; this is because a derivative of the denominator function must exist and be other than 0. Though credited to its namesake's 1696 text, this rule was probably created by John Bernoulli. FTP, name this rule from calculus used to evaluate limits that result in indeterminate ratios using derivatives; a rule named for a French mathematician.
A proof of this lemma proceeds by applying the Cauchy mean value theorem to a relevant ratio, noting that two of the difference terms vanish, then shrinking the interval.
l'Hospital's rule