<s>[INST] Calculate the limit: $ \lim\limits_{(x,y)\to(0,0)}{\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}}$ 
Calculate the limit:
$$\lim\limits_{(x,y)\to(0,0)}{\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}}
$$

I tried to change to polar coordinates like that:
\begin{align*}
\lim\limits_{(x,y)\to(0,0)}{\frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4}} &=
\lim\limits_{(x,y)\to(0,0)}{\frac{e^{-\frac{1}{x^2+y^2}}}{(x^2+y^2)^2-2x^2y^2}}\\
&=\lim\limits_{r\to0}{\frac{e^{-\frac{1}{r^2}}}{r^4(1-2\cos^2\theta \sin^2\theta)}}\\
\end{align*}
and I'm not sure how to continue from this point.
Thank you!
 [/INST] Since $1-2\cos^2 \theta \sin^2 \theta \ne 0$ the limit in polar coordinates becomes
$$
\frac{1}{1-2\cos^2 \theta \sin^2 \theta}\cdot \lim_{r\to 0}\dfrac{e^{-1/r^2}}{r^4}=0, \quad \forall \theta \in[0,2\pi)
$$
This shows that the original limit is zero.
</s>