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It's a follow up of Conjecture: $\lim\limits_{x\to 0}(x!\,x!!\,x!!!\,x!!!!\cdots )^{-1/x}\stackrel?=e$ :

A "natural" question is the following :

What is the limit of :

$$\lim_{x\to 0}\left(\frac{x!x!!!x!!!!!...}{x!!x!!!!x!!!!!!...}\right)^{-\frac{1}{x}}=?$$

For example it converge around $3/2$ but it's not exactly the value . The exact value is $e^{\gamma/(2-\gamma)}$

Now my principal interest is concentred by a another limit for example let :

$$g(x)=\left(\frac{x!x!!!x!!!!!...}{x!!x!!!!x!!!!!!...}\right)$$

Then define $a>0$ :

$$f(x)=ax$$

And :

$$h\left(x\right)=\lim_{x\to 0}\left(g\left(f\left(x\right)\right)\right)^{-\frac{1}{x}}=?$$

For $a=\pi$ the exact result is near from $3+\gamma$ see the picture below :

Approximation

What is this value $(a=\pi)$?

Miss and Mister cassoulet char
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    The same techniques people including me mentioned on that other question also work here. You'll find a rational function $r$ exists for which the limit is $e^{r(\gamma)}$. – J.G. Sep 08 '22 at 10:03
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    I get $e^{\gamma/(2-\gamma)}\approx1.500344$. – J.G. Sep 08 '22 at 10:13
  • @J.G. Hum okay can I add it in my question ? – Miss and Mister cassoulet char Sep 08 '22 at 10:21
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    You can add that value if you like. I recommend you try adapting the other question's techniques to prove it. Remember, small $x$ satisfy $x!^{\circ n}\approx1-\gamma(1-\gamma)^{n-1}x$ for $n\ge1$. – J.G. Sep 08 '22 at 10:28
  • [For any recent reader, everything below "The exact value is $e^{\gamma/(2-\gamma)}$" is an edit postdating my previous comments.] My understanding is you want $\lim_{x\to}g(ax)^{-1/x}$ now you know the case $a=1$. Hint: substitute $y=ax$ to work it out. But your exposition is very unclear; for example, I think you meant to call this limit $h(a)$, not $h(x)$. – J.G. Sep 08 '22 at 13:42

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