Find where the limit does not exist for the function Given the function: $f(x,y) = \frac{xy^4}{x^2+y^8}$, find a path where the limit does not exist at the origin.
I am having problems with this because of lot of paths go to $0$ but I know the limit does not exist.
I have tried things like $y^2$, $\sqrt(y)$ but I am getting nowhere. 
Thank you!
 A: Put $x=y^4$. Anywhere on this curve, our function is equal to $\frac{1}{2}$. So as we approach $(0,0)$ along the path $x=y^4$ (say for positive $y$), our function approaches $\frac{1}{2}$.
Put $x=-y^4$. As we approach $(0,0)$ along ths path, our function approaches $-\frac{1}{2}$.
If we want a single path such that as we approach $(0,0)$ along it, we combine the first path above with the path $y=0$. We describe the combined path geometrically. 
For $1/2\le y\le 1$, let $x=y^4$. At $y=1/2$, we reach the point $(1/2^4,1/2)$.
Travel straight down to the $x$-axis.  So we reach the point $(1/2^4,0)$. Then travel back up to $(1/2^4,1/2)$.
Now for $1/3\le y \le 1/2$, travel along $x=y^4$, until we reach the point $(1/3^4,1/3)$. Travel down to the $x$-axis, then back up again.
Continue. For $1/4\le y\le 1/3$, travel along the curve $x=y^4$, then down to the $x$-axis, back up, and so on. 
On the points of our path that are on $x=y^4$, our function is $1/2$, and on the $x$-axis, it is $0$, so if we travel towards $(0,0)$ on the complex path just described, the limit does not exist.
