= Solution
For $x\ne0$, induction gives
$$
f^{(n)}(x)=P_n(1/x)e^{-1/x^2},\qquad
P_0(t)=1,\qquad P_{n+1}(t)=2t^3P_n(t)-t^2P_n'(t),
$$
where each $P_n$ is a <polynomial>. This recurrence follows from the <product rule> and $d(1/x)/dx=-1/x^2$. Since <Gaussian decay dominates every power>,
$$
\boxed{\lim_{x\to0}f^{(n)}(x)=0\quad\text{for every integer }n\geq0.}
$$
These limits also describe the <derivatives> of the extension at zero. If $f^{(n)}(0)=0$, its next <difference quotient> is $P_n(1/h)e^{-1/h^2}/h$, which again tends to zero. Induction proves existence and <continuity> of every <derivative>, all zero there. This is a <flat function>, covered more generally by <polynomial multiples of a Gaussian flat function>.
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