I'm trying to find the leading behavior of
$$ \int_1^\infty \frac{\cos(xt)}{t}dt, \qquad x \rightarrow 0^+$$
Two applications of integration by parts gives
$$\frac{-\sin x}{x} + \frac{\cos x}{x^2} - \frac{1}{x^2} \int_1^\infty 2t^{-3} \cos(xt) dt$$
So, I believe the leading behavior is just the first two terms, but I would have to be able to show
$$ \int_1^\infty 2t^{-3} \cos(xt) dt \ll \int_1^\infty t^{-1}\cos(xt)\ dt, \qquad x \rightarrow 0^+$$
Is my reasoning correct? If so, how do I conclude this?