Let $f\left(x\right)=\underset{n\to \infty }{\lim }{\left(\frac{n^n(x+n)(x+\frac{n}{2})\mathrm{...}(x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4})\mathrm{...}(x^2+\frac{n^2}{n^2})}\right)}^{\frac{x}{n}}$ , for all $x>0$ . Then choose the correct option

$\begin{array}{l}f\left(x\right)=\underset{n\to \infty }{\lim }{\left(\frac{{\displaystyle \prod _{r=1}^n(1+\frac{1X}{n})}}{{\displaystyle \prod _{r=1}^n(1+{\left(\frac{1X}{n}\right)}^2)}}\right)}^{\frac{x}{n}}\\ =e^{x\underset{0}{\overset{1}{\int }}\ln (1+xy)dy-\ln (1+{\left(xy\right)}^2)dy}=e^{\underset{0}{\overset{x}{\int }}\ln (1-t)dt-\ln (1-t^2)dt}\\ f\text{'}\left(X\right)=f\left(x\right)\ln \left(\frac{1+x}{1+x^2}\right)\\ Forx\in (0,1)itisincrea\sin gfunction\\ f\text{'}\left(2\right)=f\left(2\right)\ln \left(\frac{3}{5}\right)<0\\ \frac{f\text{'}\left(3\right)}{f\left(3\right)}=\ln \left(\frac{2}{5}\right),\frac{f\text{'}\left(2\right)}{f\left(2\right)}=\ln \left(\frac{3}{5}\right)\end{array}$