Let f(x) = _ n → ∞ ( n ^ n (x + n)(x + (n)/(2)) . . (x + (n)/(n)) n...

Taking log both side
$lnf\left(x\right) = \lim_{n \to \infty} \dfrac{x}{n}\ln \left[\dfrac{\mathop{∏}_{r = 1}^{n}\left(x + \dfrac{1}{\dfrac{r}{n}}\right)}{\mathop{∏}_{r = 1}^{n}\left({x}^{2} + \dfrac{1}{{\left(\dfrac{r}{n}\right)}^{2}}\right)}\dfrac{1}{\mathop{∏}_{r = 1}^{n}\left(\dfrac{r}{n}\right)}\right]$
$= x\lim_{n \to \infty} \dfrac{1}{n}\mathop{∑}_{r = 1}^{n}\ln \left(\dfrac{x\dfrac{r}{n} + 1}{{\left(x\dfrac{r}{n}\right)}^{2} + 1}\right)$
Express summation in definite integration using $\dfrac{r}{n} = t \text{ and } \dfrac{1}{n} = dt$
$= x\mathop{∫}_{0}^{1}\ln \left(\dfrac{1 + tx}{1 + {t}^{2}{x}^{2}}\right)dt$
Put $tx = z$
$lnf\left(x\right) = \mathop{∫}_{0}^{x}\ln \left(\dfrac{1 + z}{1 + {z}^{2}}\right)dz$
$⇒$ $\dfrac{f′\left(x\right)}{f\left(x\right)} = \ln \left(\dfrac{1 + x}{1 + {x}^{2}}\right)$( ∵ f( x )isalwayspositive )$<br />$\forall x \in ( 0, 1 )$<br />$ln\dfrac{1 + x}{1 + {x}^{2}} > 0$<br />$⇒ \dfrac{f ' ( x )}{f( x )} > 0 \forall x \in ( 0, 1 )$<br />$asf( x ) > 0 ⇒ f ' ( x ) > 0 \forall x \in ( 0, 1 )$<br />So$f( x )$is increasing$( 0, 1 )$and${f}^{′}( 1 ) = 0 ( x > 1 ⇒ {f}^{′}( x ) < 0 )$<br />$⇒ f\left(\dfrac{1}{2}\right) < f\left(1\right), f\left(\dfrac{1}{3}\right) < f\left(\dfrac{2}{3}\right), {f}^{′}\left(2\right) < 0$<br />Also$\dfrac{{f}^{′}\left(3\right)}{f\left(3\right)} - \dfrac{{f}^{′}\left(2\right)}{f\left(2\right)} = \ln \left(\dfrac{4}{10}\right) - \ln \left(\dfrac{3}{5}\right)
$= \ln \left(\dfrac{4}{6}\right) < 0 ⇒ \dfrac{{f}^{′}\left(3\right)}{f\left(3\right)} < \dfrac{{f}^{′}\left(2\right)}{f\left(2\right)}$