I would like to get an answer or pointers to suitable material, on the following question: I know that ∫|f(x)|2dx is finite. Can we say that lim x→±∞ x*(d/dx)|f(x)|2 is zero? Are there any theorems about such limits with unknown functions that have some known properties? Basically this question arose from Griffith's introductory QM textbook which I started studying today (self study). f(x) is Ψ(x) and the author at page 16 takes the above limits to be zero "on the ground that Ψ goes to zero at ±∞". But it may go to zero asymptotically, and x goes to infinity, so we have an indeterminate form and we know only that the function's absolute value squared has a finite integral. I posted here, however, since I would like to get a more general answer on these types of limits and to be at least aware of the theory that deals with them. Thanks in advance for your precious time people!