For each positive integer n , define a function f_n on as follows :...

For each positive integer $n$, define a function $f_n$ on as follows : $f_n(x) =$\begin{cases} 0 & \text{if } x = 0 \\ \sin \frac{\pi}{2n} & \text{if } 0 < x \le \frac{1}{n} \\ \sin \frac{2\pi}{2n} & \text{if } \frac{1}{n} < x \le \frac{2}{n} \\ \sin \frac{3\pi}{2n} & \text{if } \frac{2}{n} < x \le \frac{3}{n} \\ \vdots & \vdots \\ \sin \frac{n\pi}{2n} & \text{if } \frac{n-1}{n} < x \le 1 \end{cases}$$Then the value of$\lim_{n \to \infty} \int_0^1 f_n(x) dx$ is :