Textbook Problem

Suppose that f is a positive function such that f′ is continuous.

(a) How is the graph of $y=f\left(x\right)\sin nx$ related to the graph of $y=f\left(x\right)$ ? What happens as $n\to \infty$?

(b) Make a guess as to the value of the limit

$\underset{n\to \infty }{\lim }{\displaystyle {\int }_0^{1}f\left(x\right)\sin nxdx}$

based on graphs of the integrand.

(c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since f′ is continuous, there is a constant M such that $\left|f\prime \left(x\right)\right|\le M$ for $0\le x\le 1$.]