## Norm of Derivatives of exp(-(x^2 - h^2)^(-2)) (|x| < h)

Let $\displaystyle g_h (x) = \begin{cases} e^{-\frac{1}{(x^2 - h^2)^2}, & |x| < h,\\ 0, & |x|\ge h. \end{cases} = \begin{cases} \exp(-(x^2-h^2)^{-2}), & |x| < h,\\ 0, & |x|\ge h\end{cases}.$

$\displaystyle ||g_h^{(n)}|| = \max_{|x|<h} |g_h^{(n)}(x)|$, where $\displaystyle g_h^{(n)}$ is the $\displaystyle n$th derivative of $\displaystyle g_h$.

Prove that $\displaystyle \lim_{h\to 0} ||g_h^{(n)}|| = 0$