<s>[INST] How to prove that $\frac{1}{x^4+y^4} e^{-\frac{1}{x^2+y^2}}\to0$ when $(x,y)\to (0,0) $? 
How can I calculate the limit 
  $$
\lim_{(x,y)\to (0,0)} \frac{1}{x^4+y^4} e^{-\frac{1}{x^2+y^2}}
$$
  and show that it is zero?

When switching to polar coordinates, I get:
$$
\lim_{(x,y)\to (0,0)} \frac{1}{r^4 (\cos^4 \theta+\sin^4 \theta) } e^{-\frac{1}{r^2}}
$$
but I have no idea how to show the term $\frac{1}{r^4 (\cos^4 \theta+\sin^4 \theta) }$ is bounded.
 [/INST] Another way of getting
Barry Cipra's result:
$c^4+s^4
=c^4+(1-c^2)^2
=c^4+1-2c^2+c^4
=1+2c^4-2c^2
=1-2c^2(1-c^2)
\ge 1/2
$
since
$x(1-x) \le 1/4$
for $0\le x \le 1$.
</s>