$\frac{1}{2}\underset{n\to x}{\lim }{[(1+\frac{1}{n^2})(1+\frac{2^2}{n^2})---(1+\frac{n^2}{n^2})]}^{1/n}=$

$y=\underset{n\to \infty }{\lim }{\left((1+\frac{1}{n^2})(1+\frac{2^2}{n^2})\mathrm{.......}(1+\frac{n^2}{n^2})\right)}^{1/n}$

$\log y=\underset{n\to \infty }{\lim }\frac{1}{n}$

$[\log (1+\frac{1}{n^2})+\log (1+\frac{2^2}{n^2})+\mathrm{.....}+\log (1+\frac{n^2}{n^2})]$

$\log y=\underset{n\to \infty }{\lim }\frac{1}{n}\Sigma \log (1+\frac{r^2}{n^2})$

$=\underset{0}{\overset{1}{\int }}\log (1+x^2)dx$

$={(\log (1+x^2)\int 1dx-\int \frac{d}{dx}\log (1+x^2).\int 1dx)}_0^1$

$={(x\log (1+x^2)-\int \frac{2x^2}{1+x^2}dx)}_0^1=2e^{\frac{\pi -4}{2}}$