<s>[INST] Multivariable limit using polar coordinates? 
$$\lim_{(x,y)\to(0,0)} \frac {2\sin (x^2 + y^2) + y^3}{3x^2+3y^2}$$

I am actually very new to polar coordinates so I'm a little bit confused about this part. I substitute this into this:
$\frac {2sin (r^2) + (r cos \theta)^3}{3r^2}$
But I am not even sure if it's right and what should I do after this.
Does the limit even exist? I need an explanation. Thank you!
 [/INST] \begin{align}
\lim_{(x,y)\rightarrow (0,0)}  \frac{2\sin(x^2+y^2)+y^3}{3x^2+3y^2} &= \lim_{r\rightarrow 0} \frac{2\sin(r^2)+r^3\sin^3\theta}{3r^2}\\
&=\lim_{r\rightarrow 0} \frac{2\sin(r^2)}{3r^2} + \lim_{r\rightarrow 0}\frac{r^3\sin^3\theta}{3r^2}\\
&= \frac{2}{3} + \lim_{r\rightarrow 0}\frac{r\sin^3\theta}{3}\\
&= \frac{2}{3} + 0\\
&=\frac{2}{3}
\end{align}
</s>