Summing Tiny Floors

Calculus Level 2

\[\large \displaystyle\lim_{n \to \infty} \dfrac{\lfloor x \rfloor+\lfloor 2x \rfloor+\lfloor 3x \rfloor+\cdots+\lfloor nx \rfloor}{n^2}\]

Let \(x\) be a constant real number. Find the value of the limit above in terms of \(x\).

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

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