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!

Assuming your notation means
$$\left[-x(1-F_X(x))\right]_{x=0}^{x=\infty},$$
this can be written in terms of an arbitrary parameter, say $b$, as
$$\lim\limits_{b\rightarrow\infty}\left[-x(1-F_X(x))\right]_{x=0}^{x=B}.$$
The argument of the limit simply evaluates to
$$-b(1-F_X(b))-0=-b(1-F_X(b).$$
To evaluate the original expression, we now need to calculate
$$\lim\limits_{b\rightarrow\infty}-b(1-F_X(b)),$$
recalling that $\lim\limits_{x\rightarrow\infty}F_X(x)=1$ since $F_X$ is a cumulative density function. This is done using L'Hopital's Rule.
