I have this problem and I guess I should solve it using Fourier because of the context:
let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function over $\mathbb{R}$ such that $\left|f\left(x\right)\right|\le x^4$ for all $x$.
Need to prove:
$\lim_{n\to\infty}\int_{0}^{\pi}{\frac{f\left(x\right)}{x}\cos{\left(nx\right)}dx}=0$
I can see a connection to fourier coefficients and "riemann lebesgue" maybe but I don't know how to prove it.
Let $g(x)=\frac {f(x)} x$ for $0<x<\pi$ and $g(x)=\frac {f(\pi)} {\pi}$ fo $-\pi \leq x \leq 0$. Then $g$ is integrable and Riemann Lebesgue Lemma implies that $\int_{-\pi}^{\pi} g(x) \cos (nx)dx \to 0$. Hence the result. Integrability of $g$ comes from the given inequality (which shows that $g$ is bounded).