Find limit of $f(x,y)=\frac{2x^2y}{x^4+y^2}$ as $(x,y)\to(0,0)$

Question

$$f(x,y)=\frac{2x^2y}{x^4+y^2}$$

By applying conditions such as $x=0$ and $y$ tends to $0$, or $y=mx$ and $x$ tends to $0$, we get that the limit of this function at $(0,0)$ is $0$. But how to solve it using the given definition of limit of a multivariable function

Please help

P.S- This isn't a homework question.I am solving these questions out of my own interest. In the questions that I solved before this one, I was successfully able to apply the abovementioned definition of limits and arrive at the answer. But I am unable to do so here.

Answer 1

That limit doesn't exist, because:

• if you put $y=0$, then the limit is $0$;
• if you put $y=x^2$, then the limit is $1$.