Is the Integral of f(x)/x Sin(nx) Zero as n Approaches Infinity?

[Logarythmic]

If

\frac{f(x)}{x}

is square integrable, why does

\int_a^b \frac{f(x)}{x} \sin{nx} dx \longrightarrow 0

when

n \rightarrow \infty

?

 

[OlderDan]

This is not an answer, but maybe a hint is contained in here. I think there is some theorem that will let you prove a square integrable function on a finite interval is absolutely integrable on that interval, and that demands a certain smoothness of the function. The integral of any smooth function multiplied by an infinite frequency sinusoid must be zero.