The value of $\lim _{n \rightarrow \infty}\left[\tan \dfrac{\pi}{2 n} \tan \dfrac{2 \pi}{2 n} \cdots \tan \dfrac{n \pi}{2 n}\right]^{1 / n}$ is

$\operatorname{Let} A=\lim _{n \rightarrow \infty}\left[\tan \dfrac{\pi}{2 n} \tan \dfrac{2 \pi}{2 n} \cdots \tan \dfrac{n \pi}{2 n}\right]^{1 / n}\\$

$\therefore \log A=\lim _{n \rightarrow \infty} \dfrac{1}{n}\left[\log \tan \dfrac{\pi}{2 n}+\log \tan \dfrac{2 \pi}{2 n}\right.\\$

$+\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \dfrac{1}{n} \log \tan \dfrac{\pi r}{2 n}=\int_{0}^{1} \log \tan \left(\dfrac{\pi}{2} x\right) d x\\$

$=\dfrac{2}{\pi} \int_{0}^{\pi / 2} \log \tan y d y$$I=\int_{0}^{\pi / 2} \log \tan \left(\dfrac{1}{2} \pi-y\right) d y\\$

$(\text{ by Property } \mathrm{IV})\\$

$\begin{array}{l}=\int_{0}^{\pi / 2} \log \cot y d y \\=-\int_{0}^{\pi / 2} \log \tan y d y=-I\end{array}$

$\operatorname{or} I+I=0$ or $2 I=0$ or $I=0\\$

from equation $(1), \log A=0 \therefore A=e^{0}=1$