Limit of natural log

Answer: 1

By the approximation formula
n! = n (ln n)-n+O(n) As n->∞ ; n! ~ n (ln n)

So lim (n->∞) (ln n!)^(1/n)
=lim (n->∞) [n (ln n)]^(1/n)
=lim (n->∞) [n^(1/n)][(ln n)]^(1/n)
=1*1
=1

Note :
1 lim (n->∞) n^(1/n)=1
Proof: Let L=lim (n->∞) n^(1/n)
lim (n->∞) ln L=lim (n->∞) (1/n) ln n = 0
So lim (n->∞) L = 1

2 lim (n->∞) [(ln n)]^(1/n)=1
Let L=lim (n->∞) [(ln n)]^(1/n)
ln L = lim (n->∞) (1/n) ln [(ln n)] = 0
So lim (n->∞) L = 1

1=[ln(n!)]^0 < [ln(n!)]^(1/n) < [n*ln(n)]^(1/n)
lim(n->inf) [n*ln(n)]^(1/n) = 1