Squeeze it repeatedly

Calculus Level 5

\[ \large \lim_{x\to 0} \frac{ f^{10000} (x)}{x^m} \]

Define \(f(x) = 1 -\cos(x) \) and the composite function \(f^n(x) = \underbrace{f \circ f \circ \ldots \circ f}_{n \text{ times}} \).

If the limit above equals to non-zero finite number \( \frac1{L} \) for a certain constant \(m\). Evaluate \( \log_{32} \log_2 (2L) \).


Inspiration.

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