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

\[\begin{aligned} & \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \textrm{ or } \pm\infty, \\ & \lim_{x\to c}\frac{f'(x)}{g'(x)} \textrm{ exists}, \\ & g'(x)\neq 0 \quad\forall x \in I, x\neq c, \end{aligned}\]

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

\[\begin{aligned} \lim_{n\to\infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=\ell, \end{aligned}\]

then

\[\begin{aligned} \lim_{n \to \infty} \frac{a_n}{b_n} = \ell \end{aligned}\]