Prove this identity about limit of integral with parameter Here is a question I met in an exam several days ago,I failed to give an answer that time and still feels confused now:
Suppose $f$ is monotonely increasing on $[0,1]$,prove that 
$$\lim_{y\to \infty}\int_{0}^{1}f(x)\frac{\sin(xy)}{x}dx={\pi\over 2}f(0+)$$
It's easy to see that we can assume $f(0+)=0$,but how to do the estimate next?
Should I try to find some appropriate $\delta\gt 0$ and estimate them separately?

This is a special case of Jordan's theorem, which follows from the Riemann-Lebesgue lemma, but $f(x)$ must be bounded on $[0, 1]$. You can find the proof in Apostol's Mathematical Analysis (p. 315). By the way, you cannot just assume that $f(0+) = 0$.
