I have a question. I have to check whether the limit exist. But I want to check it with a) two paths or b) polar coordinates, but it doesn't work. It is about $\lim_{(x,y)\to(0,0)}\frac{xy^2}{x^2+y^3}$. Can anyone help with to show that the limit does not exist?
Thank you
Best Answer
Let $f(x,y) := \frac{xy^2}{x^2+y^3}$. Then we have
$$f(x,y) = 1 \Leftrightarrow xy^2 = x^2+y^3 \Leftarrow x=\underbrace{\frac{y^2+\sqrt{y^4-4y^3}}2}_{=:h(y)}$$
Let $y_n:=1/n, x_n:=h(y_n)$. Then $f(x_n, y_n)\equiv 1$ and $$\lim_{n\to\infty}x_n = \lim_{n\to\infty}h(y_n) = h(\lim_{n\to\infty}x_n)= h(0) = 0\text{,}$$since $h:(-\infty,4)\to \mathbb R$ is continuous. Thus, $(x_n,y_n)\to_{n\to\infty}(0,0)$.
Also $f(0, 1/n) \equiv 0$.
So we have $\lim_{n\to\infty} f(x_n, y_n)\neq\lim_{n\to\infty} f(0, 1/n)$, and thus $\lim_{(x,y)\to(0,0)}f(x,y)$ does not exist.