I need to calculate the limit of the function: $ f(x,y) = \dfrac{2xy^3}{3x^2 + y^6 } $ at $(x,y)=(0,0)$.
When I check the orbits $y=x , y=x^{1/3} $ , I can see that such a limit doesn't exist. But if this limit doesn't exist, I should also see it in polar coordinates (I'll see that the limit is dependent on $\theta$ ). In this case I get that this limit is always $0$.
Where is my misunderstanding? Can someone help me see that the limit in polar coordinates does depend on $\theta$?
Thanks everyone!