Uniform convergence on compact sets of the derivatives of a test function

Question

For $k=1,2,...,$ let $\rho_{k}(x)=\exp\left(\dfrac{|\frac{x}{k}|^{2}}{|x|^{2}-1}\right)$ if $|x|<1$ and $ \rho_{k}(x)=0$ if $|x|\geq1$; here $x$ is in $\mathbb{R}^{n}$, $n>0$, and $|.|$ the Euclidian norm. Is it true that all derivatives of $ \rho_{k}(x) $ converge uniformly on compact sets to $0$, as $k\to\infty?$

Answer 1

The answer is no. Take $n=1.$ Then $\rho_k(0) = 1, \rho_k(1)=0.$ By the MVT,

$$-1=\rho_k(1)- \rho_k(0) = \rho_k'(c_k)\cdot 1.$$

Thus $\max_{[0,1]} |\rho_k'| \ge 1$ for all $k.$ Hence $\rho_k'$ does not converge uniformly to $0$ on $[0,1].$