Functional Equation or Limit?

Calculus Level 3

This problem is taken from Moldova National Mathematics Olympiad, 2013.

Let \(f:(0,+\infty)\rightarrow(0,+\infty)\) be differentiable and let \(F\) be a primitive of \(f\), such that \[2(F(x)-f(x))=f^2(x),\forall x\in(0,+\infty).\]

Find \(\displaystyle\lim_{x\to+\infty}\dfrac{f(x)}{x}\).

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