Answer
Limit does not exist.
Work Step by Step
We notice that if we directly substitute limits in the given function $f(x,y)=\frac{x^{4}-4y^{2}}{x^{2}+2y^{2}}$
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 $y=0$
$f(x,0)=\frac{x^{4}-4y^{2}}{x^{2}+2y^{2}}=\frac{x^{4}-0}{x^{2}+0}=x^{2}$
To evaluate limit along y-axis; put $x=0$
$f(0,y)=\frac{x^{4}-4y^{2}}{x^{2}+2y^{2}}=-2$
Thus, both limits are different along different paths therefore, limit does not exist.
