The great Ramanujan limit

Calculus Level 3

\[\large \lim_{n \to 0} \dfrac{1}{(\sqrt{n+1}+\sqrt{n})(\sqrt[4]{n+1}+\sqrt[4]{n})\cdots(\sqrt[2^{1729}]{n+1}+\sqrt[2^{1729}]{n})} = \, ?\]

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