If X is a continuous random variable and E(X) exists, does the limit as x→∞ of x[1 - F(x)] = 0? I encountered this, but so far I have neither been able to prove this, nor find a counterexample. I have tried the mathematical definition of the limit, l'Hopital's rule, integration by parts, a double integral (through expectation), and various proof scribbles, but so far, nothing has worked. Can anyone help me with this? EDIT: In this case, the function F is the CDF of X.