L'Hôpital's Rule/Weak Version

Theorem

Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$.

Let:

$\forall x \in \openint a b: \map {g'} x \ne 0$

where $g'$ denotes the derivative of $g$ with respect to $x$.

Let:

$\map f a = \map g a = 0$

Then:

$\ds \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$

provided that the second limit exists.

Proof 1

L'Hôpital's Rule/Weak Version/Proof 1

Proof 2

L'Hôpital's Rule/Weak Version/Proof 2

Examples

Example: $\dfrac {\sqrt {1 + x} - 1} x$

$\ds \lim_{x \mathop \to 0} \dfrac {\sqrt {1 + x} - 1} x = \dfrac 1 2$

Also known as

Because of variants in the rendition of L'Hôpital's name, L'Hôpital's Rule is often seen as:

L'Hospital's Rule

de L'Hôpital's rule

and so on.

Source of Name

This entry was named for Guillaume de l'Hôpital.

Historical Note

While attributed to Guillaume de l'Hôpital, who included it in his $1696$ work L'Analyse des Infiniment Petits, published anonymously, this result was in fact discovered by Johann Bernoulli in $1694$.

After L'Hôpital's death, Bernoulli claimed that most of the content of L'Analyse des Infiniment Petits, including L'Hôpital's Rule, was in fact his own work.

However, it was discovered in $1955$, on the publication of correspondence between L'Hôpital and Bernoulli that there had been an agreement between them to allow L'Hôpital to use Bernoulli's discoveries however he wanted.

Hence L'Hôpital was vindicated, and his name continues to be associated with this result.

Sources

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• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.20$: The Bernoulli Brothers
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): L'Hôpital's rule (L'Hospital's rule, de L'Hôpital's rule)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): L'Hôpital's rule (L'Hospital's rule, de L'Hôpital's rule)
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): l'Hôpital's rule
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): l'Hôpital, l'Hôpital's Rule
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): l'Hôpital's rule