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!

We have for $t>0$ that $e^t\geq\frac{t^3}3$ so 
$$\frac{\exp\left(-\frac 1{x^2+y^2}\right)}{x^4+y^4}\leq \frac 3{\frac 1{(x^2+y^2)^3}}\frac 1{x^4+y^4}=\frac{3(x^2+y^2)^3}{x^4+y^4}=3(x^2+3y^2+3x^2+y^2)=12(x^2+y^2),$$
and you are done.
