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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.