# 2014-22 Limit

For a nonnegative real number $$x$$, let $f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}$ for a positive integer $$n$$. Determine  $$\lim_{n\to\infty}f_n(x)$$.

GD Star Rating
Let f(n) be the largest integer k such that n! is divisible by $$n^k$$. Prove that $\lim_{n\to \infty} \frac{(\log n)\cdot \max_{2\le i\le n} f(i)}{n \log\log n}=1.$