An algebra problem by Patrick Bourg

Algebra Level pending

Let \(f : [0,2015] \rightarrow \mathbb R\) be the function defined as \[f(x) = \frac{1}{1+x} \quad \forall x \in [0,2015].\]

By the mean value theorem, one can show that \(\exists c_x \in (0,1)\) such that :

\[f(x) - f(0) = xf^{'}(xc_x), \quad \forall x \in [0,2015], \quad (exercise)\]

What is : \(\lim_{x \rightarrow 0} c_x\)?

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