Talk:L'Hôpital's rule

Last limit
The last limit should be 1/2. Loisel 08:16 Feb 22, 2003 (UTC)
 * Yep, fixed it. Nice catch. Minesweeper 08:37 Feb 22, 2003 (UTC)

Typo in proof?
In the case when $$|g(x)| \to +\infty$$, shouldn't the formula read

{f(x) \over g(x)} = {f(y) \over g(x)} + \left [ 1 - {g(y) \over g(x)} \right ] {f'(\xi) \over g'(\xi)} $$ instead of

{f(x) \over g(x)} = {g(y) \over g(x)} + \left [ 1 - {g(y) \over g(x)} \right ] {f'(\xi) \over g'(\xi)} $$ (the numerator of the first term of the right hand side of the equation should be f(y) instead of g(y) imho) Kind regards, Pieter Penninckx

Yes it should. Even so, somebody should finish the second proof, there's a lot more that needs to be said.

Section 1 of the proof asserts: According to Cauchy's mean value theorem there is a constant xi in c < xi < c + h such that:

f'(xi) / g'(xi) = ( f(c + h) - f(c) ) / ( g(c + h) - g(c) )

But the logic of this assertion does not seem correct to me. Cauchy's mean value theorem states:

there is a constant Xi1 in c < Xi1 < c + h   such that f'(xi1) = ( f(c+h) - f(c) ) / h

there is a constant Xi2 in c < Xi2 < c + h   such that g'(xi2) = ( g(c+h) - g(c) ) / h

So certainly,

f'(xi1) / g'(xi2) = ( f(c+h) - f(c) ) / ( g(c+h) - g(c) )

However, you cannot assume that xi1 = xi2 !

"Hence Cauchy's mean value theorem ...it states that xi1 = xi2! And it's not proved like that." ~P. Y. from NTHU

I'm not saying the assertion is wrong, but I think the proof needs improvement.


 * Indeed, Cauchy's mean theorem CAN'T be derived that way. Instead it uses Rolle's theorem, see (http://en.wikipedia.org/wiki/Rolle%27s_theorem) and (http://en.wikipedia.org/wiki/Mean_value_theorem).

(http://planetmath.org/?op=getobj&from=objects&id=7611). The main text needs to be corrected.
 * The proof of "With the indeterminate form infinity over infinity" is simply wrong. The correct proof can be found here

Too strong requirement in overview?
Hello,

We were touched that $$ \lim_{x\to c}{f'(x) \over g'(x)} = A, A \in \mathbb{R}^* $$ requirement holds only for open interval (a,b) containing c (or with $$b=\infty$$ or $$a=-\infty$$)

Incorrect application of L'Hopital's rule in example image
The example below the infobox shows $$\lim_{x\rightarrow{0}}\frac{\sin{x}}{-0.5x} = -\frac{1}{2}\lim_{x\rightarrow{0}}\frac{\sin{x}}{x}$$. However, L'Hopital's rule cannot be applied to this limit, as the proof of the derivative of $$\sin{x}$$ depends on knowing precisely this limit.

Proof: $$ \frac{d}{dx}\sin{x}=\lim_{h\rightarrow{0}}\frac{\sin{(x+h)}-\sin{x}}{h}=\lim_{h\rightarrow{0}}\frac{\sin{x}\cos{h}+\cos{x}\sin{h}-\sin{x}}{h}=\cos{x}\lim_{h\rightarrow{0}}\frac{\sin{h}}{h}+\sin{x}\lim_{h\rightarrow{0}}\frac{\cos{x}-1}{h}, \lim_{h\rightarrow{0}}\frac{\sin{h}}{h}=1, \lim_{h\rightarrow{0}}\frac{\cos{x}-1}{h}=0 \therefore\frac{d}{dx}\sin{x}=\cos{x} $$

I think this should be changed to a different example.

Singularities421 (talk) 21:54, 21 March 2021 (UTC)


 * L'Hopital's rule is not used to prove sin(x) ~ x. Valery Zapolodov (talk) 12:14, 10 November 2022 (UTC)