L'Hopital's rule for more than one variable?

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SUMMARY

The discussion addresses the existence of an analog to l'Hopital's rule for functions of multiple variables. It confirms that while traditional l'Hopital's rule applies to single-variable limits, directional derivatives can be utilized for multivariable limits. The example provided demonstrates the limit of sin(x² + y²)/(x² + y²) as (x,y) approaches (0,0) using directional derivatives, yielding a limit of 1. The polar form of the limit is also shown to arrive at the same conclusion.

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Is there an analog to l'Hopital's rule for functions of more than one variable? Or am I stuck using [itex]\epsilon[/itex] [itex]\delta[/itex] proofs and the squeeze theorem? Those also depend on me knowing the value of the limit beforehand which can be tricky in itself.
 
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lim f/g=lim {[D_n]f}/{[D_n]g}
where D_n is the directional derivative in the direction of the limit
ie
[D_n]f=dx fx+dy fy
where fx and fy are partial derivatives
example
lim (x,y)->0 sin(x^2+y^2)/(x^2+y^2)=lim cos(x^2+y^2)(2xdx+2ydy)/(2xdx+2ydy)=1
or in polar form
lim r->0 sin(r^2)/(r^2)=lim cos(r^2)(2r.dr)/(2r.dr)=1
 

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