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L’Hôpital’s Rule

Let $a,L\in\mathbb{R}\cup\{-\infty,\infty\}$ be extended real numbers, and let $I$ be an open interval containing $a\,(\text{if }a\in\mathbb{R})$ or having $a$ as an endpoint$\,(\text{if }a=\pm\infty)$

Let $f,g:I\setminus\{a\}\to\mathbb{R}$ be differential functions on $I\setminus\{a\}$ such that $g'(x)\neq 0 \quad \text{for all} \quad x\in I\setminus\{a\}$

Suppose that one of the following conditions holds:

1. $\lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0$
2. $\lim_{x \to a} \lvert g(x) \rvert= \infty$

If the limit of the raio of their derivatives exists, that is,

Then the limit of ratio of the functions exists and is given by:

Proof of the $0/0$ case

First suppose that $a,L\in\mathbb{R}$

we may define $f(a)=g(a)=0$ with these definitions, f and g are continuous at a

By definitions of a limit,

Take $x$ such that

By the Cauchy Mean Value Theorem, there exists $c$ strictly between $a$ and $x$ such that

Since $f(a)=g(a)=0$,

Since $c$ lies between $a$ and $x$,

Therefore,

Hence,

Proof of other case

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