In this limit i don't quite sure that the answer is correct can somebody please help me $$\lim_{(x,y)\to\ (0,0)} \frac{2x^2y+y^3}{x^2+y^2}=$$ $$\lim_{r\to\ 0} \frac{2r^2\cos^2\theta r\sin\theta+r^3\sin^3\theta}{r^2}=$$ $$\lim_{r\to\ 0}\frac{r^3(2\cos^2\theta\sin\theta+\sin^3\theta)}{r^2}=$$ $$\lim_{r\to\ 0}2r\cos^2\theta\sin\theta+\sin^3\theta=$$ My answer is that the limit doesn't exist but i don't know for sure
Thanks in advance
Be careful with parenthesis. The last line should be $$\lim_{r\to\ 0}r(2\cos^2\theta\sin\theta+\sin^3\theta).$$ Now for any fixed $\theta$, $(2\cos^2\theta\sin\theta+\sin^3\theta)$ is constant with respect to $r$. So what is the above limit?