Answer
$\ln 3$
Work Step by Step
The polar-coordinates are defined as: $x= r \cos \theta , y = r \sin \theta$ and $r^2=x^2+y^2$
$ \lim\limits_{(x,y) \to (0,0) } f(x,y)=\lim\limits_{(x,y) \to (0,0) } \ln (\dfrac{3x^2-x^2y^2+3y^2}{x^2+y^2}$
or, $=\lim\limits_{r \to 0} \ln (\dfrac{ 3r^2 \cos^2 \theta- r^2 \cos^2 \theta r^2 \sin^2 \theta+3r^2 \sin^2 \theta}{r^2})$
or, $=\lim\limits_{r \to 0} \ln (3\cos^2 \theta-r^2 \cos ^2 \theta \times \sin ^2 \theta+3 \sin ^2 \theta)$
or, $=\lim\limits_{r \to 0} \ln [(3) (\cos^2 \theta + \sin ^2 \theta)]$
or, $=\ln 3$
