<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] You can also use polar coordinate by letting $x = r\cos \theta$ and $y = r\sin\theta$ and the limit becomes $$\lim_{r\to 0} \frac{r^5 \cos \theta \sin^4\theta}{r^4\cos^4\theta + r^2\cos^2\theta + r^4\sin^4\theta} = \lim_{r\to 0} r^3\left(\frac{\cos\theta\sin^4\theta}{r^2\cos^4\theta+\cos^2\theta + r^2\sin^4\theta}\right) = 0$$
Today's lesson is : Don't trust Wolframalpha entirely
</s>