Limit as n approaches infinity of (n!)^(1/n)/(n)

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A different approach is to use Stirlings formula: n! ≈ √(2 π n) (n/e)^n . This formula is asymptotically correct .
One only needs to show that √(2 π n)^(1/n) -> 1 as n->∞ , which is not hard if you look at the logarithm of the expression.