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Let $g(n) = n - \big \lfloor \frac{n}{2} \big \rfloor+ \big \lfloor \frac{n}{3} \big \rfloor - \big \lfloor \frac{n}{4} \big \rfloor + \cdots.$

Evaluate $\displaystyle \lim_{n \to \infty}\frac{g(n)}{n}.$

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