L'Hôpital's rule

In calculus, l'Hôpital's rule is a theorem relating the limit of the quotient of two functions to the limit of the quotient of their derivatives. It is most useful for evaluating indeterminate limits.

Statement

Let $c\in ℝ\cup \{-\infty ,\infty \}$,i.e. an extended real number. and let $f$ and $g$ be functions differentiable over an open interval containing $c$. Assuming $$\underset{x\to c}{\lim }\frac{fx}{gx}=L,$$ then l'Hôpital's rule states that either $$\underset{x\to c}{\lim }fx=\underset{x\to c}{\lim }gx=0$$ or $$\underset{x\to c}{\lim }|fx|=\underset{x\to c}{\lim }|gx|=\infty$$ implies $$\underset{x\to c}{\lim }\frac{f^{\prime }x}{g^{\prime }x}=\underset{x\to c}{\lim }\frac{fx}{gx}=L.$$