L'Hôpital in several variables I am wondering if there is a multidimensional analog of l'Hôpital's rule for functions of several variables.
I have searched online for a while and have found people that argue both sides. One said it was possible by replacing the derivative with the directional derivative but I didn't quite understand.
 A: An essential ingredient in the proof of L'Hôpital's rule is Rolle's theorem for differentiable functions  $f:{\mathbb R}\to{\mathbb R}$ that guarantees the existence of a $\tau\in\ ]a,b[\ $ with $f'(\tau)=0$ if $f(a)=f(b)=0$. As a consequence this rule is not applicable even for complex functions of a real variable, let alone for functions of several variables. 
Consider in this regard the following example (a similar example can be found in Rudin's "Principles of Mathematical Analysis"): Let
$$f(t):=t\ ,\quad g(t):=te^{-i/t}\qquad(t>0)\ .$$
Then one has $\lim_{t\to0+}f(t)=\lim_{t\to0+}g(t)=0$, so that L'Hôpital's rule would be called for. One computes $f'(t)\equiv1$, $g'(t)=\bigl(1+{i\over t}\bigr)e^{-i/t}$.
Since $\bigl|e^{-i/t}\bigr|=1$ we therefore have
$$\lim_{t\to0+}{f'(t)\over g'(t)}=\lim_{t\to 0+}{t\over t+i}\ e^{-i/t}=0\ .$$
So L'Hôpital's rule would tell us that $$\lim_{t\to0+}{f(t)\over g(t)}=0\ ;$$
but in reality this limit does not exist, as $t\mapsto f(t)/g(t)=e^{i/t}$ goes around the unit circle an infinite number of times when $t\to0+$.
A: Below are some references. I previously posted these in December 2005 (URLs just below), but as I haven't kept up with this topic, it's possible that something relevant may have been published since 2005.
http://groups.google.com/group/alt.math.undergrad/msg/eb8efd19eebab8f0
http://mathforum.org/kb/message.jspa?messageID=4143759
[1] Eugen Dobrescu and Ioan Siclovan, Considerations on functions of two variables (Romanian), Analele Universitatii Timisoara Seria Stiinte Matematica-Fizica 3 (1965), 109-121. [MR 34 #5998; Zbl 166.31502]
[2] A. I. Fine and S. Kass, Indeterminate forms for multi-place functions, Annales Polonici Mathematici 18 (1966), 59-64. [MR 32 #7680; Zbl 137.03603]
[3] Ira Rosenholtz, A topological mean value theorem for the plane, American Mathematical Monthly 98 (1991), 149-154. [MR 91m:26014; Zbl 741.26003]
[4] Tadeusz Wazewski, Une généralisation des théoremes sur les accroissements finis au cas des espaces abstraits. Applications, Bull. Int. Acad. Polon. Sci. Lett., Cl. Sci. Math. Natur., Ser. A 1949, 183-185. [MR 12,508a; Zbl 41.23301]
[5] Tadeusz Wazewski, Une généralisation des théorèmes sur les accroissements finis au cas des espaces de Banach et application à la généralisation du théorème de l'Hôpital, Ann. Soc. Polon. de Math. 24 (1951), 132-147. [MR 15,717g; Zbl 52.11302]
[6] Tadeusz Wazewski, Une modification due théorème de l'Hôpital liée au problème du prolongement des intégrales des équations différentielles, Annales Polonici Mathematici 1 (1954), 1-12. [MR 16,118e; Zbl 56.11402]
[7] William H. Young, On indeterminate forms, Proceedings of the London Mathematical Society (2) 8 (1910), 40-76. [JFM 40.0334.01]
from p. 71 of Young's paper: "We now pass to one or two generalisations to more than one variable. It has commonly, but erroneously, been supposed, that such generalisations did not exist."
A: The responsible thing is to point out that there is nothing really useful here.
I screwed up!! 
First, let $f(x,y) = x + y + x^2 + y^2 + x^3,$ while $g(x,y) = x + y + x^2 + y^2.$ There are directional limits (1) along each line approaching the origin, and these all agree. However, along the parametrized curve 
$$ x = \frac{-1}{2} + \frac{\cos t}{\sqrt 2}, \; \;  y = \frac{-1}{2} + \frac{\sin t}{\sqrt 2},$$ as $t$ approaches $\frac{\pi}{4}$ we are approaching the origin. One may confirm that, along this curve, $g$ is always 0, while $f$ is nonzero except at the origin itself. So the ratio is not even defined, and there is no limit. 
In future I promise not to write down things as theorems if I just made them up. 
Sigh. 
