Fact of Factorials

Let \(f(n) : \mathbb{Z}^+ \to \mathbb{Z}\) be a function which gives the number of trailing zeros in \(n!\).

Then find,

\[\lim_{n\to \infty} \frac{f(n)}{n}\]

Details and Assumptions

  • For example, if \(n=3 \), then \(3! = 6 \), so \(f(3) = 0\); if \(n=7\), then \(7! = 5040\), so \(f(7) = 1 \)
Image Credit: Wikimedia Factorial Interpolation
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