Let $f(x)=\lim_{n\to\infty}\left(\frac{n^{n}(x+n)\left(x+\frac{n}{2}\right)\cdots\left(x+\frac{n}{n}\right)}{n!\left(x^{2}+n^{2}\right)\left(x^{2}+\frac{n^{2}}{4}\right)\cdots\left(x^{2}+\frac{n^{2}}{n^{2}}\right)}\right)^{\frac{x}{n}}$, for all $x>0$. Then

Let f(x)=lim_(n→∞)((nⁿ(x+n)(x+n/2)·s(x+n/n))/(n!(x²+n²)(x²+n²/4)·s(x²+n²/n²)))^(x/n), for all x>0. Then

$f(x)=\lim_{n\to\infty}\left(\dfrac{\text{product of }(x+n/k)}{\dfrac{n!}{n^{n}}\ \text{product of }(x^{2}+n^{2}/k^{2})}\right)^{\frac{x}{n}}$

$\text{To find: which of } (1)\text{--}(4) \text{ are correct}$

$L_n=\dfrac{n^{n}\prod_{k=1}^{n}(x+\frac{n}{k})}{n!\prod_{k=1}^{n}(x^{2}+\frac{n^{2}}{k^{2}})}$

$\ln f=\lim_{n\to\infty}\dfrac{x}{n}\ln L_n$

$\prod_{k=1}^{n}\left(x+\dfrac{n}{k}\right)=\dfrac{n^{n}}{n!}\prod_{k=1}^{n}\left(1+\dfrac{xk}{n}\right)$

$\prod_{k=1}^{n}\left(x^{2}+\dfrac{n^{2}}{k^{2}}\right)=\dfrac{n^{2n}}{(n!)^{2}}\prod_{k=1}^{n}\left(1+\dfrac{x^{2}k^{2}}{n^{2}}\right)$

$L_n=\dfrac{\prod_{k=1}^{n}\left(1+\dfrac{xk}{n}\right)}{\prod_{k=1}^{n}\left(1+\dfrac{x^{2}k^{2}}{n^{2}}\right)}$

$\ln f=x\left[\int_0^1\ln(1+xt)\,dt-\int_0^1\ln(1+x^{2}t^{2})\,dt\right]$

$\int_0^1\ln(1+xt)\,dt=\dfrac{1+x}{x}\ln(1+x)-1$

$\int_0^1\ln(1+x^{2}t^{2})\,dt=\ln(1+x^{2})-2+\dfrac{2}{x}\arctan x$

$\ln f(x)=(1+x)\ln(1+x)-x\ln(1+x^{2})+x-2\arctan x$

$\dfrac{f'(x)}{f(x)}=\dfrac{d}{dx}\ln f(x)=\ln\dfrac{1+x}{1+x^{2}}$

$\dfrac{1+x}{1+x^{2}}\ge 1 \iff 0\le x\le 1 \;\Rightarrow\; f'>0 \text{ on }(0,1),\ f'<0 \text{ on }(1,\infty)$

$f\left(\dfrac{1}{3}\right)\le f\left(\dfrac{2}{3}\right) \;\Rightarrow\; (2)\ \text{TRUE}$

$f\left(\dfrac{1}{2}\right)<f(1) \;\Rightarrow\; (1)\ \text{FALSE}$

$\dfrac{f'(2)}{f(2)}=\ln\dfrac{3}{5}<0 \;\Rightarrow\; f'(2)<0 \;\Rightarrow\; (3)\ \text{TRUE}$

$\ln\dfrac{4}{10}=\ln 0.4<\ln 0.6=\ln\dfrac{3}{5} \;\Rightarrow\; (4)\ \text{FALSE}$

$\text{Correct: } (2)\ \text{and}\ (3)$