Answer
Does not exist
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}$$
Consider the line $y=mx$ that passes through $(0,0)$:
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}&=\lim _{x \rightarrow0} \frac{x^{4}-m^{4}x^4}{x^{4}+x^{2} m^{2}x^2+m^{4}x^4}\\
&=\lim _{x \rightarrow0} \frac{1-m^{4}}{1+ m^{2}+m^{4}}\\
&=\frac{1-m^{4}}{1+ m^{2}+m^{4}}
\end{align*}
Since the limit depends on $m$, then $\lim _{(x, y) \rightarrow(0,0)} \dfrac{x^{4}-y^{4}}{x^{4}+x^{2} y^{2}+y^{4}}$ does not exist.
