Limit of a composite function!

Calculus Level 4

$\large \lim_{x\to0} \left[\lim_{n\to\infty} 2^{2n} \left(1 - f^n(x) \right)\right]$

For $$f: [0,1] \rightarrow \mathbb R$$ and $$f(x) = \sqrt{\frac{1+x}2}$$, find the limit above.

Clarification: $$f^n(x) = \underbrace{f \circ f \circ f \circ \ldots \circ f}_{n \text{ times}} (x)$$.

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