$\lim_{R\rightarrow +\infty}\int_{-1}^{1}\frac{\sin(2\pi R t)}{t}dt=\frac{1}{\pi}$

Question

As suggested by Reuns in in this question. It is interesting that the problem there boils down to the following limit:

$$\lim_{R\rightarrow +\infty}\int_{-1}^{1}\frac{\sin(2\pi R t)}{t}dt=\frac{1}{\pi}.$$

Anyone has an idea.

Thanks so much.

Answer 1

Let $2\pi Rt=x$, then $$I =\lim_{R\rightarrow\infty} \int_{-2\pi R}^{2\pi R} \frac{\sin x}{x} dx= \int_{-\infty}^{\infty} \frac{\sin x}{x} dx= 2 \int_{0}^{\infty} \frac{\sin x}{x} dx=2\frac{\pi}{2}=\pi$$