What is the value of this term and why is it not included in the given passage?

[St41n]

I found this passage in a book:
http://img8.imageshack.us/img8/1452/75717730.jpg
where Φ(x) is the c.d.f. of the normal distribution.
However, using integration by parts I also get this term which is not included in the passage:

[ e^{itx} \phi(x) ]^{+ \infty}_{- \infty}
where i is the imaginary unit and \phi(x) is the normal p.d.f.
So,
e^{itx} \phi(x) = \frac{1}{\sqrt{2\pi}} \exp \left( -0.5x^2 + itx \right)
So, is this term:
\left[ \frac{1}{\sqrt{2\pi}} \exp \left( -0.5x^2 + itx \right) \right]^{+ \infty}_{- \infty}
equal to zero and why? It is not clearly evident to me

 

[HallsofIvy]

Yes, as x go to infinity,
e^{-x^2}
goes to 0 very quickly. Quickly enough so that even
e^{-x^2}e^{x} goes to 0 as x goes to positive infinity.

 

[St41n]

I kept doing the same fault, but I figured it out now, thanks