For every integrable function $f \colon \mathbb{R} \to \mathbb{C}$ I have to calculate the limit 
$$\lim\_{|\lambda| \to +\infty} \int\_{\mathbb{R}}f(x)|\sin(\lambda x)|dx 
$$

We know that this limit exists because it is bounded by the integral of $f(x)$. From the lemma of Riemann-Lebesgue, we know that 
$$\lim\_{|\lambda| \to +\infty} \int\_{\mathbb{R}}f(x)\sin(\lambda x)dx = 0$$
So I assume that the limit that I need to calculate is bigger than zero, and is somewhere close to $$\int\_{\mathbb{R}}f(x)dx$$
but I have no idea how I can prove this.