Infinity /47

let

$\large{L=\displaystyle \lim_{n\to \infty} \sqrt [ n ]{ \sin { \left( \frac { \pi }{ 4n } \right) } \sin { \left( \frac { 2\pi }{ 4n } \right) ......... } \sin { \left( \frac { \left( \frac { n }{ 47 } -1 \right) \pi }{ 4n } \right) } } }$

$\large{\ln (L)=\displaystyle \lim_{n\to \infty} \frac{1}{n} \displaystyle \sum^{\frac{n}{47}-1}_{r=1} \ln(\sin (\frac{r \pi}{4n}))}$

Reimann integration method

$\large{\ln L=\displaystyle \int^{\frac{1}{47}}_{0} \ln(\sin(\frac{\pi}{4}x))dx}$

Frankly i used wolfram alpha after this step

$\large{\ln L=-0.108}$

$\large{L=e^{-0.108}=\boxed{0.897}}$