Q:

A functional analysis question with respect to ODE

If $f:[0,T] \times \mathbb{R} \to \mathbb{R} $ has continuous derivatives, let $x_k(\cdot)$ denote the solution of the following equation:
$$\left\{\begin{array}{ll}
\dot{x}=f(t,x)\sin(kt)\\
x(0)=0
\end{array} ,\right. 
$$
then does the limit $\lim_{k\to\infty}x_k(t)$ exist?  Why? 
There is a hint that I should use weak convergence but I have no idea about it. 

A:

You may recast the problem as a fixed point equation:
$$x(t)=\int_0^t f(s,x(s)) \sin(ks) ds.$$
(This is the initial idea behind Picard iteration, which is the proof technique for the famous Picard-Lindelof theorem.) 
Suppose $x$ is any continuous function.  Then what is the limit of the right side as $k \to \infty$?  You may want to look up the Riemann-Lebesgue lemma.