Check if the point $(0,0)$ is a removable discontinuity for the function:
$$f(x,y)=\frac{x^2y^4}{x^4+y^8} \sin\left(\frac 1 x \right)\cos\left(\frac 1 y \right)$$
So to prove this I assume I would need to find the limit of the function at the point? Even, so I'm lost as to how I would do that, given the $\sin$ and $\cos$ functions which mess things up and that I'm working with two variables. Some help would be appreciated.
No, it is not removable. The limit at $(0,0)$ taken along the line $x=y^2$ doesn't exist, since then we have$$\frac{y^8}{2y^8}\sin\left(\frac1{y^2}\right)\cos\left(\frac1y\right)\left(=\frac12\sin\left(\frac1{y^2}\right)\cos\left(\frac1y\right)\right).$$