Unfamiliar Limit

Calculus Level 5

\[\color{red}{\mathfrak{L}}=\displaystyle\lim_{n\to\infty}\left(\ln\left(\left(\dfrac{n^2-1^2}{n^2}\right)\cdot\left(\dfrac{n^2-2^2}{n^2}\right)\cdots\left(\dfrac{2n-1}{n^2}\right)\right)^{1/n}\right)\]

If \(\color{red}{\mathfrak{L}}\) can be expressed in the form \( \color{red}{\ln\left(\varphi \large{e}^{\alpha}\right)^{\gamma}}\) for \(\varphi,\alpha,\gamma\in \mathbb Z-\{1\}\), then:

\[\large (\varphi+\alpha+\gamma)!=\ ?\]

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