Let $l>0$, $f(x)$ is continuous on $[-l,l]$ and differentiable at $x=0$. Please use Riemann lemma to prove that
$$\begin{equation} \lim_{n\rightarrow\infty}{\frac{1}{\pi}\int_{-l}^{l}{f(x)\frac{\sin{nx}}{x}}dx}=f(0) \end{equation}$$
I am sorry for being stupid but I am stuck at this question for 2 hours with no clue of how to even start. Please help me......
Hint: $\frac 1 {\pi} \int_{-l} ^{l} \frac {sin (nx)} x dx\to 1$ as $n \to \infty$ (as seen by making the substitution $y=nx$). Hence it is enough to show that $\int_{-l}^{l} (f(x)-f(0)) \frac {sin (nx)} x dx \to 0$. Let $g(x)= \frac {f(x)-f(0)} x$ for $x \neq 0$ and $g(0)=f'(0)$. Apply Riemann Lebesgue Lemma to this function to finish the proof.