Solution
= Solution
The <pointwise limit> is zero, including at $x=0$. Differentiation gives
$$
\frac{d}{dx}(xe^{-nx})=e^{-nx}(1-nx),
$$
so the maximum on $[0,\infty)$ is attained at $x=1/n$ and equals $1/(en)$. Therefore
$$
\boxed{\sup_{x\geq0}xe^{-nx}=\frac1{en}\to0,}
$$
and \b[the convergence is uniform], even though the domain is unbounded.