L’Hôpital’s rule:
for functions \(f\) and \(g\) which are differentiable on \(I \backslash\{c\}\) ,
where \(I\) is an open interval containing \(c\), if
then
\[\begin{aligned} \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. \end{aligned}\]Stolz-Cesàro theorem:
for real-number sequences \((a_n) _{n \ge 1}\) and \((b_n)_{n \ge 1}\),
assume \(b_n\) is stictly increasing and unbounded, and
then
\[\begin{aligned} \lim_{n \to \infty} \frac{a_n}{b_n} = \ell \end{aligned}\]