Is 0 divided by a non-zero indeterminate equal to 0. Consider the following solution: 
$$ \lim\limits_{(x, y) \to (0, 0)} \dfrac{xy^4}{x^4+y^4}$$
Divide both numerator and denominator by $y^4$
$$ =  \lim\limits_{(x, y) \to (0, 0)} \dfrac{x}{\left(\dfrac{x}{y}\right)^4+1}$$
$$ =  \dfrac{ \lim\limits_{(x, y) \to (0, 0)}x}{ \lim\limits_{(x, y) \to (0, 0)}\left(\dfrac{x}{y}\right)^4+1}$$
The numerator is 0 and the denominator is non-zero, hence the limit is 0
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Are you satisfied with this solution, if not, why? 
Edit: Thank You for the help guys. 
My mistake was looking at the final limit and not paying attention to how I got to that stage, "by dividing both numerator and denominator by y^4", that step itself is not allowed because y can be 0. 
The above solution will be complete if I include another case where $y=0$, because when you divide by $y^4$, you are implicitly stating that y is not 0. 
 A: What you have shown is that assuming $y\neq 0$ the limit is zero but it doesn't suffice.
What matter is that for $y\neq 0$
$$ \dfrac{|x|}{\left(\dfrac{x}{y}\right)^4+1} \le |x| \to 0$$
and since for $y=0 \implies \dfrac{xy^4}{x^4+y^4}=0$ the proof is complete.
A: There is an easy way:
\begin{align*}
\left|\dfrac{xy^{4}}{x^{4}+y^{4}}\right|&=|x|\dfrac{y^{4}}{x^{4}+y^{4}}\\
&\leq|x|\\
&\rightarrow 0.
\end{align*}
A: One thing that has not been adressed by the existing answers is that even if you restrict your reasoning to the cases where $y\neq 0$, it's still wrong. To write something like
$$\lim_{(x,y)\to (0,0)}\frac{f(x,y)}{g(x,y)}=\frac{\lim_{(x,y)\to (0,0)} f(x,y)}{\lim_{(x,y)\to (0,0)}g(x,y)}$$
you have to make sure that $g$ is non-zero around the identity and that both $\lim_{(x,y)\to (0,0)} f(x,y)$ and $\lim_{(x,y)\to (0,0)} g(x,y)$ exist, and that the latter is not $0$. But here $g(x,y)=\left(\frac{x}{y}\right)^4+1$ does not have a limit when $(x,y)\to (0,0)$, so you can't actually write something like
$$ \lim\limits_{(x, y) \to (0, 0)} \dfrac{x}{\left(\dfrac{x}{y}\right)^4+1}=  \dfrac{ \lim\limits_{(x, y) \to (0, 0)}x}{ \lim\limits_{(x, y) \to (0, 0)}\left(\dfrac{x}{y}\right)^4+1}$$
because the right-hand side does not make sense.
