# Generalizing l'Hopital's Rule

Mandelbroth

Out of curiosity, is anyone aware of a kind of generalization for l'Hopital's Rule from analysis for differentiable maps between differentiable manifolds? I'm having trouble formulating if I could do it or not, because (as far as I know), if I have ##f,g:M\to N##, with ##f## and ##g## differentiable and ##M## and ##N## differentiable, ##f(x)/g(x)## is not, in general, defined.

Again, I don't know if it can be generalized. Ideas are certainly welcome, since I'll probably be stuck thinking about it until I prove something does work or doesn't work.

Gold Member
Maybe there's something like:

$\lim_{x\rightarrow 0} \frac{||f(x)||}{||g(x)||} = \frac{||\nabla f(0)||}{||\nabla g(0)||}$

But it's just a guess, I didn't confirm this.
Anyway, you can't define division between two functions unless they are scalars, obviously.

Edit: Well, you can define some sort of division but it wouldn't be like the case we know from calculus.

Gold Member
It seems to me that the next identity should be fulfilled.

If ##f(x_1,\cdots , x_m)=(f_1,\cdots, f_n)## and ##g(x_1,\cdots,x_m)=(g_1,\cdots,g_n)##, then:

##||f||/||g|| = \sqrt{f^2_1+\cdots+f^2_n}/\sqrt{g^2_1+\cdots + g^2_n}##

Now it seems to me to be plausible that: