TITLE: Two-variable limit, quotient of polynomials
QUESTION [7 upvotes]: I'm trying to evaluate the following limit,
$$ \lim_{(x,y)\to(0,0)} \frac{x^3-y^2}{x^2-y} $$
which I think it doesn't exist, since for the curve $\alpha :[0,1]\to \mathbb R^2$, $\alpha(t) = (t, t^2)$ it isn't well defined, and if the limit exists it is because for every continuous curve $\gamma:[0,1]\to \mathbb R^2$ such that $\gamma(0) = (0,0)$ and $\gamma(t) \neq (0,0)$ for all $t\neq 0$, and $\lim_{t\downarrow 0} f(\gamma(t)) = 0.$  Is that correct? How would you evaluate the limit? Any hint? Thanks in advance.

REPLY [3 votes]: If $f(x,y)=(x^3-y^2)/(x^2-y)$ then $f(t,t^2+t)=t(t^2+t+1)$ while $f(t,t^2+t^3)=t^3+2t^2+t-1.$ So as $t \to 0$ one of them goes to $0$ and the other goes to $-1$.