Answer
$0$
Work Step by Step
Given: $\lim\limits_{(x,y) \to (0,0)}f(x,y)=\frac{xy^{4}}{x^{4}+y^{4}}$
Divide both numerator and denominator by $y^{4}$.
$=\lim\limits_{(x,y) \to (0,0)}\frac{x}{(\frac{x}{y})^{4}+1}$
$=\frac{\lim\limits_{(x,y) \to (0,0)}x}{\lim\limits_{(x,y) \to (0,0)}(\frac{x}{y})^{4}+1}$
$=\frac{0}{0+1}$
$=0$
Hence, the limit converges to $0$.
