For each positive integer \( n \), define a function \( f_{n} \) on \( [0,1] \) as follows :
\[
...
For each positive integer \( n \), define a function \( f_{n} \) on \( [0,1] \) as follows :
\[
f_{n}(x)=\left\{\begin{array}{clc}
0 & \text { if } & x=0 \\
\sin \frac{\pi}{2 n} & \text { if } & 0x \leq \frac{1}{n} \\
\sin \frac{2 \pi}{2 n} & \text { if } & \frac{1}{n}x \leq \frac{2}{n} \\
\sin \frac{3 \pi}{2 n} & \text { if } & \frac{2}{n}x \leq \frac{3}{n} \\
\sin \frac{n \pi}{2 n} & \text { if } & \frac{n-1}{n}x \leq 1
\end{array}\right.
\]
Then the value of \( \lim _{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) d x \) is :
(a) \( \pi \)
(b) \( \frac{\pi}{2} \)
(c) \( \frac{1}{\pi} \)
(d) \( \frac{2}{\pi} \)
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