Difficult Multivariate limit problem

[Scriabin11]

Homework Statement

I am supposed to show whether the limit:

lim (x,y)----> (0,0) of yln(x^2 + y^2) exists or doesn't.

Homework Equations

The Attempt at a Solution

I've tried numerous paths, but what it seems to come down to is showing that the y factor goes to zero quicker than the ln(x^2 +y^2) factor. And I don't believe l'hospital's rule is applicable.

 

[╔(σ_σ)╝]

L'hospitals rule is applicable since you end of with inderterminate forms when you approach x and y separately.

 

[HallsofIvy]

In polar coordinates, $y ln(x^2+ y^2)$ becomes $r sin(\theta) ln(r^2)= (2 r ln(r))sin(\theta)$. That will go to 0 independently of y if and only if $\lim_{r\to 0} r ln(r)= 0$. Is that true?