Functional Equation or Limit?

Calculus Level 3

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

Let f:(0,+)(0,+)f:(0,+\infty)\rightarrow(0,+\infty) be differentiable and let FF be a primitive of ff, such that 2(F(x)f(x))=f2(x),x(0,+).2(F(x)-f(x))=f^2(x),\forall x\in(0,+\infty).

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

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