For $0 < a < b$, find $$ \lim_{n\rightarrow \infty}\int_{a}^{b} \frac{\sin (nx)}{x} dx.$$
My attempt : $$\int_{a}^{b} \frac{\sin (nx)}{x} dx= \sin x\int_{a}^{b}\frac{1}{x}dx -\int_{a}^{b}\cos x\cdot \log x dx $$ after that I'm not able to proceed further.
Pliz help me. Any hints/solution?
Thanks