Square integrable/vanish at infinity?

[kharranger]

Let f be a differentiable complex valued function on R. If f is square integrable, then it is not the case that f(x) must approach zero at infinity. counterexample: f(x)=x^2 exp(-x^8 sin^2(20x)).
If I also require that the derivative of f be square integrable, is that enough to guarantee that lim x-> +/- infinity of f(x) is zero?
Thanks,

KH

 

[mathwonk]

i think square integrable means the quare of the aBSOLUTE VAlue is finite.