A Problem from NIMO April Fun Round 2016

Algebra Level 5

For all real numbers \(x\) not equal to \(2\), let \(\Gamma(x) = \frac{1}{2−x}\). Define the iterative function \( \Gamma^{{n+1}} (x) = \Gamma^{(n) } \left( \Gamma (x) \right) \).

If \( \Gamma^{n} \left( \frac{6}{29} \right) \) can be represented as \( \frac{a_n}{b_n} \) for relatively prime positive integers \(m, n \), determine

\[ \lim_{n\rightarrow \infty } | a_n - b_n |. \]

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