<s>[INST] Find the Limit of $‎\prod‎_{n=1}^{‎\infty}\frac{(1+‎\frac{1}{n}‎)^n}{(1+‎\frac{1}{n+x})^{n+x}}$ ‎Consider the following productions‎
‎$$‎‎‎‎‎\prod‎_{n=1}^{‎\infty}\frac{1+‎\frac{1}{n}‎}{1+‎\frac{1}{n+x}}$$
and
$$‎‎‎‎‎\prod‎_{n=1}^{‎\infty}\frac{(1+‎\frac{1}{n}‎)^n}{(1+‎\frac{1}{n+x})^{n+x}}$$
I know that the above are convergent.
Can anyone find the limit of these products?
‎
 [/INST] For the first one, rewrite the product: 
\begin{equation*}
\begin{split}
\prod_{n=1}^{\infty} \frac{(n+1)(n+x)}{n(n+x+1)}& = \frac{(1+1)(1+x)}{1(1+x+1)} \times\frac{(2+1)(2+x)}{2(2+x+1)} \times \frac{(3+1)(3+x)}{3(3+x+1)}\times ...\\
& = \frac{2(1+x)}{1(2+x)} \times\frac{3(2+x)}{2(3+x)} \times \frac{4(3+x)}{3(4+x)}\times ...\\
& = (1+x).
\end{split}
\end{equation*}
For the second one, rewrite the product as
\begin{equation*}
\begin{split}
\prod_{n=1}^{\infty} \frac{(n+1)^n(n+x)^{n+x}}{n^n(n+x+1)^{n+x}} & = \frac{(1+1)^1(1+x)^{1+x}}{1^1(1+x+1)^{1+x}} \times \frac{(2+1)^2(2+x)^{2+x}}{2^2(2+x+1)^{2+x}}\times \frac{(3+1)^3(3+x)^{3+x}}{3^3(3+x+1)^{3+x}}\times ...\\
& = \frac{2(1+x)^{1+x}}{(2+x)^{1+x}} \times \frac{3^2(2+x)^{2+x}}{2^2(3+x)^{2+x}}\times \frac{4^3(3+x)^{3+x}}{3^3(4+x)^{3+x}}\times ...\\
& = (1+x)^{1+x} \times \frac{(2+x)}{2}\times \frac{(3+x)}{3}\times ...\\
& = (1+x)^{1+x}\prod_{n=2}^{\infty}\frac{n+x}{n}.
\end{split}
\end{equation*}
The product is equal to zero for $x<0$, it is equal to $1$ for $x=0$, and for $x>0$ the product is divergent.
</s>