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

1. Feb 26, 2009

### m00npirate

Is there an analog to l'Hopital's rule for functions of more than one variable? Or am I stuck using $\epsilon$ $\delta$ proofs and the squeeze theorem? Those also depend on me knowing the value of the limit beforehand which can be tricky in itself.

Last edited: Feb 26, 2009
2. Feb 26, 2009

### lurflurf

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