L'Hôpital's rule for $\mathbb R ^n \to \mathbb R $ functions. I want to know if there exists a generalization of  L'Hopital rule in $n$ dimensions? For example, let us consider this problem.
There it is just said that we should take separate  path and see if they will end with same  number,but can't we generalize L'Hopital rule and partial derivatives in $n$ dimensions? Just a little hint will help me too much.
 A: l'Hopitals rule is, in its essence, an extension or corollary from the (extended) mean value theorem. There is no such simple mean value theorem in the vector case. 
Your example
$$\frac{4x^2-y^2}{2x-y}=\frac{(2x-y)(2x+y)}{2x-y}$$ 
works because you can cancel the common factor. But take any example where $(1,2)$ is still the common zero but now without common factor, like
$$\frac{x+1-y}{5x-1-2y},$$
then on almost every line through $(1,2)$ you get a limit by l'Hopitals rule, but the limit is different for every direction.
A: Some thoughts.
Let $f,g\colon \mathbb R ^2\to \mathbb R$ be $C^2$ functions over $\mathbb R ^2$ such that $f(0,0)=g(0,0)=0$. Suppose that $g$ is injective in a neighborood of $( 0,0 )$, so that $\frac{f(x,y )}{g(x,y)}$ is well defined.
Given a couple $(x,y)$ sufficiently near the origin, it is true that $$\dfrac{f(x,y)}{g(x,y)}=\dfrac{\partial _x f(\tau x,\tau y)x+\partial _y f(\tau x,\tau y)y}{\partial _x f(\tau x,\tau y)x+\partial _y f(\tau x,\tau y)y},\qquad \tau \in (0,1),$$
(this is Cauchy's theorem applied to the functions $f \circ \gamma$ and $g\circ \gamma$, $\gamma(t)=(tx,ty)$).
So, if $$\ell = \{\dfrac{\partial _x f(x,y) \cdot x + \partial _y f(x,y) \cdot y}{\partial _x g(x,y) \cdot x + \partial _y g(x,y) \cdot y}\}_{(x,y)\to (0,0)}$$
exists, then the limit of $\frac{f}{g}$ also exists and it is equal to $\ell$.

EDIT. After some thought, I'm not sure that the existence of this limit, implies what I was claiming. I think some extra hypotesis are required, but I can't see which are, right now. Actually I'm trying to prove it only using $C^2$ continuity. 

EDIT2: Proof of the claim.
Suppose $\ell$ exists. By continuity of the function:$$h(x,y)=\dfrac{\partial _x f(x,y) \cdot x + \partial _y f(x,y) \cdot y}{\partial _x g(x,y) \cdot x + \partial _y g(x,y) \cdot y},$$
hence local uniform continuity, given $\varepsilon >0$ we can find $\delta>0$ such that $$|h(x',y')-h(x,y)|<\varepsilon$$
if $|(x,y)|<\delta$ and $|(x',y')|<\delta$. 
So if we take $|(x,y)|<\delta$, nothing that $$h(\tau x,\tau y)=\dfrac{\partial _x f(\tau x,\tau y)x+\partial _y f(\tau x,\tau y)y}{\partial _x f(\tau x,\tau y)x+\partial _y f(\tau x,\tau y)y},$$
we have $$|h(\tau x,\tau y)-\ell|\leq |h(\tau x, \tau y)-h(x,y)|+|h( x , y) -\ell|<2\varepsilon$$
(if we take at most a smaller $\delta$).

The hypotesis on $g$ can be replaced with the much weaker: $g(x,y)\neq g(0,0)$ in a neighborood of $g$. However, I don't know how much the last one can be easy to prove. In the special case of radial functions $g(x^2+y^2)$, this can be reconducted to the crucial hypothesis for using D.H. rule: $\frac{\text{d}g}{\text d r}\neq 0$.
