#### Answer

limits does not exist.

#### Work Step by Step

We notice that if we directly substitute limits in the given function $f(x,y)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}$
Then $f(0,0)=\frac{0}{0}$
We get intermediate form. Therefore, we will calculate limit of function in following way.
To evaluate limit along x-axis; put $y=0$
$f(x,0)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}=\frac{5(0)^{4}cos^{2}x}{x^{4}+0}=0$
To evaluate limit along y-axis; put $x=0$
$f(0,y)=\frac{5y^{4}cos^{2}x}{x^{4}+y^{4}}=5$
For a limit to exist, all the paths must converge to the same point.
Hence, both the limits are different and follow different paths, thus, given limits does not exist.