<s>[INST] Calculate $\lim\limits_{(x,y)\to (0,0)} \dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}}$ Calculate, $$\lim\limits_{(x,y)\to (0,0)} \dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}},$$ if there exist.
My attempt:
I have tried several paths, for instance: $x=0$, $y=0$, $y=x^m$. In all the cases I got that the limit is $0$. But I couldn't figure out how to prove it. Any suggestion?
 [/INST] Assuming $y<1$ and using polar coordinates we have
$$\dfrac{x^4}{(x^2+y^4)\sqrt{x^2+y^2}} \le \dfrac{x^4}{(x^2+y^2)\sqrt{x^2+y^2}} =r \sin^4\theta \to 0$$
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