limn!^1nn

By using the ratio test, we can simplify the expression as follows:

lim[(n+1)!^(1/(n+1)) * n^n / (n!)^(1/n)] / (n+1)

= lim[(n+1)^((n+1)/n) * n^(1/n)] / (n+1)

= lim[(n+1)^(1/n) * n^(1/n)] / (n+1) * (n+1)^(1/n)

= lim[(1 + 1/n)^(n/n) * (n^(1/n) / (n+1)^(1/n))] * lim[(n+1)^(1/n)]

As n goes to infinity, the first limit goes to e^(-1) and the second limit goes to 1. Therefore, the overall limit is:

lim[(n!)^(1/n)]/n = e^(-1) * 1 = e^(-1)