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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