Let $f(x)=\lim _{n \rightarrow \infty}\left(\frac{n^n(x+n)\left(x+\frac{n}{2}\right) \ldots\left(x+\frac{n}{n}\right)}{n!\left(x^2+n^2\right)\left(x^2+\frac{n^2}{4}\right) \ldots\left(x^2+\frac{n^2}{n^2}\right)}\right)^{\frac{x}{n}}$, for all $x>0$. Then
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