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?
Best Answer
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 $x=y^2$: $$ \frac{x^2+y^3}{xy-x+y^2} = \frac{y^4+y^3}{y^3-y^2+y^2} = \frac{y^3(y+1)}{y^3} = y+1 \to 1. $$