As you can find a strange limit in my best post here on MSE I have tried to find a similar strange limit it start with the definition :
$$f_n\left(x\right)=\left(x!\left(\frac{1}{x}\right)!\left(x^{2}\right)!\left(\frac{1}{x^{2}}\right)!\left(x^{3}\right)!\left(\frac{1}{x^{3}}\right)!\left(x^{4}\right)!\left(\frac{1}{x^{4}}\right)!\left(x^{5}\right)!\left(\frac{1}{x^{5}}\right)!...\left(x^{n}\right)!\left(\frac{1}{x^{n}}\right)!\right)^{-1}$$
Unfortunately trying some limit exponent as $1^{\infty}$ or other else I cannot find an interesting limit . But it's not the main subject here .
The function $f(x)$ behaves like $e^{-c(x-1)^2}$ and we have better result introducing some exponent as we can define $g_n\left(x\right)=f_n\left(x\right)^{\frac{x^{2}}{\ln\left(x+1\right)}}$
Question :
Does it have some interesting pattern in maths or physics ? Does $\lim_{n\to\infty}f_n(x)$ admits a closed form or have a good upper bound ?
Ps : here $x!=\Gamma(x+1)$
As n approached infinity it's not to difficult to examine the behavior of the function. Notice how $x!$ approaches 1 as x gets close to 0. using this, $(x^k)!(\frac{1}{x^k})!$ would then approach positive infinity for values of x much greater than one and less than one. Combining this with the product and the reciprocal, you can see that all values not close to 1 will go to zero and as n approaches infinity, you will be left with a spike at x=1 of height 1