L'Hôpital's rule in the context of Johann Bernoulli

L'Hôpital's rule (/ˌloʊpiːˈtɑːl/ loh-pee-TAHL), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to de l'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.

L'Hôpital's rule states that for functions ${\displaystyle f}$ and ${\displaystyle g}$ which are defined on an open interval ${\displaystyle I}$ and differentiable on ${\displaystyle I\setminus \{c\}}$ for a (possibly infinite) accumulation point ${\displaystyle c}$ of ${\displaystyle I}$, if ${\displaystyle \lim \limits _{x\to c}f(x)=\lim \limits _{x\to c}g(x)=0}$ or ${\displaystyle \pm \infty ,}$ and ${\displaystyle g'(x)\neq 0}$ for all ${\displaystyle x}$ in ${\displaystyle I\setminus \{c\}}$, and ${\displaystyle \lim \limits _{x\to c}{\frac {f'(x)}{g'(x)}}}$ exists, then