TITLE: find the limit of sequence $\ln(n) ^{1/n}$
QUESTION [5 upvotes]: I'm not sure if I can use the squeeze theorem??? 
I did the following, not sure if it's correct 
$$1^{1/n} < \ln(n)^{1/n} < n^{1/n}$$ for n greater than or equal to 3 
 and as  $$\lim_{n\to\infty} 1^{1/n} = 1\qquad \text{and}\qquad
\lim_{n\to\infty} n^{1/n} = 1\qquad \text{(it's a fact)}$$
 then $$\lim_{n\to\infty} \ln(n)^{1/n} = 1$$ ??????

REPLY [3 votes]: Yes it is correct.
As an alternative:
$$(\ln n)^{\frac1n}=e^{\frac1n\ln (\ln n)}\to e^0=1$$
NOTE
in general when you have expression like $A^B$ it can be useful transform in the equivalent expression:
$$A^B=e^{ln A^B}=e^{B\ln A}$$