# Rational limit on two variables approaching the origin

I am dealing with the following limit: $$\lim_{\left (x,y\right )\to \left (0,0\right )}\frac{x^2+y^3}{xy-x+y^2}$$. Wolfram Alpha says it does not exist: http://www.wolframalpha.com/input/?i=lim_%7B(x,y)%5Cto+(0,0)%7D%5Cfrac%7Bx%5E2%2By%5E3%7D%7Bxy-x%2By%5E2%7D

So, I was trying to find two paths approaching the origin which give me different limits when composed with the rational function. Of course, paths like $$\left (t,0\right )$$, $$\left (0,t\right )$$ give the limit $$0$$.

Some other paths I used: $$\left (t^a\cos t,t^b\sin t\right )$$ where $$a,b\in \mathbb{N}_0$$, $$\left (\left (1-e^t\right )\frac{t^2}{1-t},t\right )$$, $$\left (t^3\cos t,t\cos t\right )$$ and some others. All of them gave me the limit $$0$$.

How would you solve the exercise?

For path $$x=0$$ we have: $$\frac{x^2+y^3}{xy-x+y^2} = \frac{y^3}{y^2} = y \to 0,$$
on the other hand, for $$x=y^2$$: $$\frac{x^2+y^3}{xy-x+y^2} = \frac{y^4+y^3}{y^3-y^2+y^2} = \frac{y^3(y+1)}{y^3} = y+1 \to 1.$$