Solving Multi-Limits: x to 0, y to 0

[supercali]

Homework Statement

solve
$${\lim }\limits_{\scriptstyle x \to 0 \hfill \atop <br /> \scriptstyle y \to 0 \hfill} (xy\ln \left( {x^2 + y^2 } \right))$$

Homework Equations

$$y_{1}=x$$
$$y_2=1/x$$

The Attempt at a Solution

when using $$y_{1}=x$$ we get that the limit is zero
when using this $$y_{2}=1/x$$ we get that the limit is infinity
thus the limit does not ! exist.
i don't know if it is ok to use these 2 paths i some how have a feeling that the second one is wrong
thanks for the help

 

[Vid]

Convert it to polar and let r->0. You get that the limit is zero. Also, the whole idea of using a path is to show the limits do not exist by showing that if you approach the point 0,0 along different paths then you get different limits. y = 1/x never approaches (0,0)