# l’Hôpital’s rule – 로피탈의 정리

In calculus, l’Hôpital’s rule (also spelled l’Hospital) uses derivatives to help compute limits with indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to a determinate form, allowing easy computation of the limit. The rule is named after the 17th-century French mathematician Guillaume de l’Hôpital, who published the rule in his book Analyse des infiniment petits pour l’intelligence des lignes courbes (literal translation: Analysis of the infinitely small to understand curves) (1696), the first book to be written on differential calculus.

The rule is believed to be the work of Johann Bernoulli since l’Hôpital, a nobleman, paid Bernoulli a retainer of 300₣ per year to keep him updated on developments in calculus and to solve problems he had. (Moreover, the two signed a contract allowing l’Hôpital to use Bernoulli’s discoveries in any way he wished.)[1] Among these problems was that of limits of indeterminate forms. When l’Hôpital published his book, he gave due credit to Bernoulli and, not wishing to take credit for any of the mathematics in the book, he published the work anonymously. Bernoulli, who was known for being extremely jealous, claimed to be the author of the entire work, and until recently, it was believed to be so. Nevertheless, the rule was named for l’Hôpital, who never claimed to have invented it in the first place[2].

The Stolz-Cesàro theorem is a similar result involving limits of sequences, and using finite difference operators rather than derivatives.

Introduction

In simple cases, l’Hôpital’s rule states that for functions f(x) and g(x), if:

$\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=0,$

or:

$\lim_{x \to c}f(x)=\lim_{x \to c}g(x)=\pm\infty,$

then:

$\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}$

where the prime (‘) denotes the derivative.

Among other requirements, for this rule to hold, the limit $\lim_{x\to c}\frac{f'(x)}{g'(x)}$ must exist.
Other requirements are detailed below, in the formal statement.

Proofs of l’Hôpital’s rule
Proof by Cauchy’s mean value theorem

The most common proof of l’Hôpital’s rule uses Cauchy’s mean value theorem.

With the indeterminate form 0 over 0

The case when $f(x) \to 0, g(x) \to 0$

First, we expand continuously (or redefine) f(x) and g(x) by 0 for x = c. This doesn’t change the limit since the limit doesn’t depend on the value in the point (by definition).

According to Cauchy’s mean value theorem there is a constant ξ in c < ξ < c + h such that:

$\frac{f'(\xi)}{g'(\xi)} = \frac{f(c + h) - f(c)}{g(c + h) - g(c)}$

Since f(c) = g(c) = 0,

$\frac{f'(\xi)}{g'(\xi)} = \frac{f(c + h)}{g(c + h)}$

If $h \to 0$ then $\xi \to c$ and

$\lim_{x\to c}\frac{f'(x)}{g'(x)} = \lim_{\xi\to c}\frac{f'(\xi)}{g'(\xi)} = \lim_{h\to 0}\frac{f(c+h)}{g(c+h)} = \lim_{x\to c}\frac{f(x)}{g(x)}$

With the indeterminate form infinity over infinity

The case when $|g(x)| \to +\infty$

Let x < y < x + h. Then using Cauchy’s mean value theorem:

$\frac{f'(\xi)}{g'(\xi)} = \frac{f(x) - f(y)}{g(x) - g(y)}$

We rewrite that in the form
$\frac{f(x)}{g(x)} = \frac{f(y)}{g(x)} + \left [ 1 - \frac{g(y)}{g(x)} \right ] \frac{f'(\xi)}{g'(\xi)}$

and then by the discussion of the two cases

$\begin{cases} \lim_{x \to c}\frac{f'(x)}{g'(x)} = B \in \mathbb{R} \\ \lim_{x \to c}\frac{f'(x)}{g'(x)} = \pm \infty \end{cases}$

we show that the limit of f(x)/g(x) tends to the same when $x \to c$ and $h \to 0$.

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