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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.

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    http://www.math.umn.edu/~zyf/mc/Aside2011-11-01

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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.

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    Interesting and much more complicated than expected. This brings a related question to mind. Suppose $f,g$ are analytic at $(x_0,y_0)$, then could I reduce the limit of $f/g$ to the limit of the first few nonzero terms of their Taylor series, or something similar in this manner?2011-11-02
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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 $t\to0+$.

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    I realize that just straight up taking the derivative of the top and bottom does not work, but I feel like there could be _some_ differential operator which does the trick.2011-11-01
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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."