<s>[INST] Wrong Wolfram Alpha result for $\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}$? 
I'm trying to solve this limit:
  $$
\lim_{(x,y)\to(0,0)}\frac{xy^4}{x^4+x^2+y^4}
$$


Here's my attempt:
$$0 \le |\frac{xy^4}{x^4+x^2+y^4} - 0| = \frac{|x|y^4}{x^4+x^2+y^4},$$ and since $x^4+x^2 \ge0$ then $\frac{y^4}{x^4+x^2+y^4} \le 1$ so 
$$
\frac{|x|y^4}{x^4+x^2+y^4} \le |x|,$$ so 
$$
0 \le \lim_{(x,y)\to(0,0)}|\frac{xy^4}{x^4+x^2+y^4} - 0| \le \lim_{(x,y)\to(0,0)} |x| = 0,
$$ and using the squeeze theorem the limit is $0$. 
But if I input the limit in wolfram alpha, it says that the limit doesn't exist. Here is the link to the limit in Wolfram Alpha.
 [/INST] For 
\begin{align*}
\lim_{(x,y)\rightarrow(0,0)}\dfrac{xy^{4}}{x^{4}+x^{2}+y^{2}},
\end{align*}
one does the following step which is similar to your technique:
\begin{align*}
\left|\dfrac{xy^{4}}{x^{4}+x^{2}+y^{2}}\right|\leq|xy^{2}|\cdot\dfrac{y^{2}}{x^{4}+x^{2}+y^{2}}\leq|x|\cdot y^{2}\rightarrow 0
\end{align*}
as $(x,y)\rightarrow(0,0)$.
</s>