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