For each positive integer $n$, define a function $f_n$ on $[0,1]$ as follows:
$$f_n(x) = \left\{ \begin{array}{ccc} 0 & \text{if} & x = 0 \\ \sin\frac{\pi}{2n} & \text{if} & 0 < x \leq \frac{1}{n} \\ \sin\frac{2\pi}{2n} & \text{if} & \frac{1}{n} < x \leq \frac{2}{n} \\ \sin\frac{3\pi}{2n} & \text{if} & \frac{2}{n} < x \leq \frac{3}{n} \\ \vdots & \vdots & \vdots \\ \sin\frac{n\pi}{2n} & \text{if} & \frac{n-1}{n} < x \leq 1 \end{array} \right.$$
Then, the value of $\lim_{n \rightarrow \infty} \int_0^1 f_n(x) dx$ is\\
(A) $\pi$\\
(B) 1\\
(C) $\frac{1}{\pi}$\\
(D) $\frac{2}{\pi}$