Find $\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} $ I'm trying to find $$\lim\limits_{(x,y) \to (0,0)} \frac{e^{-\frac{1}{x^2+y^2}}}{x^4+y^4} .$$ After I tried couple of algebraic manipulation, I decided to use the polaric method. I choose $x=r\cos \theta $ , $y=r\sin \theta$, and $r= \sqrt{x^2+y^2}$, so I get $$\lim\limits_{r \to 0} \frac{e^{-\frac{1}{r^2}}}{r^4\cos^4 \theta+r^4 \sin^4 \theta } $$ What do I do from here? Thanks a lot!

You are almost done!

$$\lim_{r\to 0}\frac{e^{-\frac{1}{r^2}}}{r^4(\cos^4\theta+\sin^4\theta)}=\frac 1{\cos^4\theta+\sin^4\theta}\lim_{r\to 0}\frac{e^{-\frac{1}{r^2}}}{r^4}.$$ For the last limit you can use L'Hopital's rule.