The unusual infinity

Calculus Level 5

\[\lim_{x\to\infty} x^{-x^2} \bigg[ (x+1)\left(x+\frac12\right)\left(x+\frac1{2^2} \right)\ldots \left(x + \frac1{2^{x-1}} \right) \bigg]^x = \ ? \]

Give your answer to 3 decimal places.

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