Explanation

(c)
$\lim_{n \rightarrow \infty}{z_{1}z_{2}\ldots z_{n}} = \lim_{n \rightarrow \infty}\left( \cos\frac{\alpha}{n^{2}} + i\sin\frac{\alpha}{n^{2}} \right) \times \left( \cos\frac{2\alpha}{n^{2}} + i\sin\frac{2\alpha}{n^{2}} \right)\ldots\left( \cos\frac{n\alpha}{n^{2}} + i\sin\frac{n\alpha}{n^{2}} \right)$
$= \lim_{n \rightarrow \infty}\left\lbrack {cols}\left\{ \frac{\alpha}{n^{2}}(1 + 2 + 3 + ... + n) \right\} + i\sin\left\{ \frac{\alpha}{n^{2}}(1 + 2 + 3 + ... + n) \right\} \right\rbrack$
$= \lim_{n \rightarrow \infty}\left\lbrack \cos{\frac{\alpha n(n + 1)}{2n^{2}} + i\sin\frac{\alpha n(n + 1)}{2n^{2}}} \right\rbrack$
$= \cos{\frac{\alpha}{2} + i\sin{\frac{\alpha}{2} = e^{\frac{i\alpha}{2}}}}$