Answer
$0$
Work Step by Step
Conversion of polar co-ordinates $(r, \theta)$ are: $x=r \cos \theta$ and $y= r \sin \theta$
Given: $\lim\limits_{(x,y) \to(0,0)}(x^2+y^2) \ln (x^2+y^2)$
This implies
$=\lim\limits_{r \to0}(r^2 \cos^2 \theta+r^2 \sin^2 \theta) \ln (r^2 \cos^2 \theta+r^2 \sin^2 \theta)$
$=\lim\limits_{r \to 0}r^2 (\cos^2 \theta+\sin^2 \theta) \ln (r^2 \cos^2 \theta+r^2 \sin^2 \theta)$
$=\lim\limits_{r \to 0} r^2 \ln r^2$
$:=\lim\limits_{r \to 0}\dfrac{2/r}{-2/r^3}$
$=0$
