Why must $\Psi (x,t)$ go to zero faster than $\frac{1}{\sqrt{|x|}}$ as $|x|$ goes to $\infty$?
According to Griffiths' Introduction to Quantum Mechanics, it must. I don't understand why, and this is in his footnote (while talking about normalizability), so there's no explanation as to why this must be so.