# Need some hint with limit with multiple variables

The limit $$\lim_{(x,y)\to(0,0)} \frac{x^{1/3}y^2}{x+y^3}$$ does exist since when we set $y=0$, $x=0$, and $x=y$ (it all equals to $0$). So now we need to evaluate the limit. The L'Hospital rule does not apply with multiple variables, so I am stuck of how to approach it. I understand that I need to somehow manipulate it, but I have no idea how. Any hints?

• Does it? What about the limits along the paths $x = y^3$ and $x = 2y^3$? – stochasticboy321 Oct 6 '17 at 3:56
• It doesn't exist. – IntegrateThis Oct 6 '17 at 3:56

Hint: try the curve $x=y^3$ as the trajectory. To express the fact that $(x,y)\to(0,0)$ along this trajectory, substitute $x=y^3$ and then find the limit as you let $y\to0$ (along this curve).