Limit Driller #1

$L = \displaystyle \lim_{n \to \infty} \left[ \prod_{r=1}^{n}\tan \left(\dfrac{r\pi}{2n} \right) \right]^{\frac{1}{n}}$

$\ln L = \displaystyle \lim_{n \to \infty} \dfrac{1}{n} \left[\sum_{r=1}^{n} \ln \left(\tan\left(\dfrac{r\pi}{2n} \right)\right)\right]$

This is a Riemann Integral.

$\ln L = \displaystyle \int_{0}^{1} \ln(\tan\left(\dfrac{\pi x}{2}\right))dx = 0$

$\Rightarrow L = e^0 = \boxed{1}$

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Note

• $\displaystyle \int_{0}^{\frac{\pi}{2}} \ln (\tan x) dx = \int_{0}^{\frac{\pi}{2}} \ln (\cot x) dx = 0$

Limit = $\displaystyle e^{\int\limits_0^1 \log(tan(\pi x/2)) dx}$

= $e^0$

= $\boxed 0$