Can l'Hopital's Rule Be Generalized for Differentiable Maps Between Manifolds?

[Mandelbroth]

I had a wild thought.

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.

 

[MathematicalPhysicist]

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.

 

[MathematicalPhysicist]

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:
$\lim_{x\to 0} ||f||/||g|| = \frac{\sqrt{(grad \ f_1)^2+\cdots + (grad \ f_n )^2}}{\sqrt{(grad \ g_1)^2+\cdots + (grad \ g_n )^2}}(at \ x=0)$