## Calculus 8th Edition

$0$
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 $0$.