Question

Let $f(x) = \lim_{n\to\infty} \left( \frac{n^n (x+n)(x+\frac{n}{2})...(x+\frac{n}{n})}{n!(x^2+n^2)(x^2+\frac{n^2}{4})...(x^2+\frac{n^2}{n^2})} \right)^{\frac{1}{n}}$

Answer 1

$\displaystyle f(x)=\exp\Biggl\{\int_0^1\ln\frac{1+xt}{1+x^2t^2}\,dt\Biggr\}$

Answer 2

Rewrite each product by factoring out the dominant $n$–terms and take logarithms. Express the sums as Riemann integrals in the limit $n\to\infty$. After cancellation and using Stirling’s formula, the limit becomes
$\ln f(x)=\int_0^1\ln\frac{1+xt}{1+x^2t^2}\,dt$. Exponentiating gives the answer.