Infinitely nested inverse

Calculus Level 3

Let \(f : [-1, 1] \to [0, \pi]\) be defined as \(f(x) = \arccos(x)\). Define \(f_{n}(x)\) as \(\underbrace{f \circ f \circ f \circ \cdots \circ f}_{\text{n times}}\) if it's defined.

For some \(x \in [-1, 1] \), \(\displaystyle \lim_{n \to \infty}{f_{n}(x)} = A\), a real number. Evaluate \(A\) to \(3\) decimal places.

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