Evaluating the following expression Why is the value of the following expression equal to 0? I have a feeling that I need to apply L'Hopital's rule, but I do not know where.
$[-x(1-F_X(x)]\Big|_0^{\infty}$, where $F_X(x)$ is the cumulative distribution function of the random variable $X$. 
Thanks!

The limit evaluates to $0$ for $x=0$. We need to use L'Hopital's Rule to evaluate the limit at $\infty$. 
$lim_{x→\infty}[-x(1-F_X(x)]$ = $lim_{x→\infty}[-x]/lim_{x→\infty}\frac{1}{1-F_X(x)}$ = $lim_{x→\infty}[-1]/lim_{x→\infty}\frac{f_X(x)}{[1-F_X(x)]^2}$ = $lim_{x→\infty}\frac{[1-F_X(x)]^2}{f_X(x)}$ = $\frac{0}{\infty}$ = $0$.
