limit and uniform convergence of sequence of function I have a sequence of function $g_n$:
$$ g_n(x) = xe^{-nx}, \qquad \text{for   } x \in [0,\infty )
$$
I need to find the limit,
and determine the uniform convergnece of the sequence.
I think the limit is 0, but having trouble determining whether the convergence is uniform.

A reasonable conjecture for the (pointwise) limit is $g(x)=0$. So let's see if $g_n\to g$ uniformly on $[0,\infty)$. But this means
$$
\max_{x\in[0,\infty)}|g_n(x)-g(x)|\to 0\text{ as }n\to\infty.
$$
Just a little calculus reveals that the maximum occurs at $x={1\over n}$ so
$$
\max_{x\in[0,\infty)}|g_n(x)-g(x)|=g_n(1/n)={1\over e\,n}\to 0\text{ as }n\to\infty.
$$
Thus, $g_n(x)\to g(x)=0$ uniformly on $[0,\infty)$.
