Todd and Rodd

Calculus Level 5

\[\begin{eqnarray} f_n(x) &=& \tan(f_{n-1}(x)) \quad, \quad f_1 = \tan(x) \\ g_n(x) &=& \arctan(g_{n-1}(x)) \quad,\quad g_1 = \arctan(x) \end{eqnarray}\]

Given the two recurrence relations above for \(n=2,3,4,\ldots \). Evaluate

\[\large \lim_{x\to0} \dfrac{f_{2016}(x) - g_{2016}(x) }{x^3}. \]

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