How to explain this limit exists? I am trying to show the $\lim_{(x,y)\to (0,0)}\frac{x^4y}{x^2-y^2}$ exists or doesn't exist and if it exists what it is. I have chosen a path $y = x - kx^4$ and am able to show that this goes to $\frac{1}{2k}$. I believe it is appropriate to say this limit exists using this path but is it true to say that the $\lim_{(x,y)\to (0,0)} f(x,y)$ is equal to $\frac{1}{2k}$? The problem is I am not exactly how to explain why and what exactly this means. I have seen something proved like this before but I really don't understand why this proves the limit exists. Any help in understanding why would be appreciated.

When you compute the limit along the path $y=x-kx^4$ and come to the conclusion that it is $\frac{1}{2k}$, you are  showing that the limit does not exist. If it existed, you would obtain the same limit along every possible path.
