\[\large f\left( \dfrac xy \right) = f( x) - f( y) \]

Let \( f(x)\) be defined for all real \(x > 1\) such that it satisfies the above functional equation for all real \(x\) and \(y\) and that \(f(e) =1\). Which is of the following options is correct?

- \(P: f \left( x \right) \) is bounded
- \(Q: f\left( \dfrac 1x \right) \to 0\) as \(x\to 0\)
- \(R: xf\left( x \right) \to 1\) as \(x \to 0\)
- \(S: f\left( x \right) =\ln x\)

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