Prove point-wise and uniform convergence of function (Carothers 10.9.f)

Question

I am trying to complete exercise 10.9.f in Carothers Real Analysis. The ask is to provide a formal proof that $nxe^{-nx}$ converges pointwise and determine if it uniformly converges (if not, find a subinterval that does).

I propose the limit is 0, but I'm am not sure how to define the $n\geq N$ that will make $|nxe^{-nx}|<\epsilon$ true.

Answer 1

The limit is $0$ for $x \geq 0$ but it is $-\infty$ for $x<0$. The convergence is not uniform on $(0,\infty)$ because $nxe^{-nx}=e^{-1}$ when $x=\frac 1 n$.

Hint for showing that the limit is $0$ for $x>0$: Use the inequality $e^{nx} >\frac {n^{2}x^{2}} 2$.