Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$ 
Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$.

$f$ is continuously differentiable.
I'm told this can be done using basic calculus. It's difficult for me to see where I should begin. I'd like some hints.

If the function $f$ is continuous the limit is $0$. Just notice that $f$ is uniformly continuous, and that on every small interval $[2k\pi/n,(2k+2)\pi/n]$ the function $sin(nx)$ has integral $0$ while $f$ is close to a constant. So the integral will be close to $0$.
In general consider the piecewise constant function obtained replacing $f$ with its mean value on every such small intervals... and prove that the integral of the difference (between $f$ and the piecewise constant function) goes to zero. In practice you want to find $f_n$ such that 
$$
  \left\vert \int_a^b f(x) \sin(nx)\right\vert \le \left\vert \int_a^b (f(x)-f_n(x)) \sin(nx)\right\vert + \left\vert \int_a^b f_n(x) \sin(nx)\right\vert = \int_a^b \lvert f(x)-f_n(x)\rvert < \varepsilon.
$$
