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}