How to prove that lim n → ∞ n n = 1 ?

Jamir Summers

Jamir Summers

Answered

2022-11-25

How to prove that lim n n n = 1 ?

Answer & Explanation

Dillan Foley

Dillan Foley

Expert

2022-11-26Added 9 answers

Rewrite as
n n = n 1 / n = e ln ( n ) / n
Now, you can write that
lim n n n = lim n e ln ( n ) / n = e lim n ln ( n ) / n
Looking at the exponent, you have (using L'Hopital's Rule)
lim n ln ( n ) n = lim n 1 / n 1 = lim n 1 n = 0
Therefore, you have
lim n n n = e 0 = 1
DinamisGr

DinamisGr

Expert

2022-11-27Added 3 answers

Observe that
n n = n 1 / n = e 1 n log ( n ) .
Since e : R R is a continuous function, taking the limit gives
lim n n n = lim n e 1 n log ( n ) = e lim n 1 n log ( n ) .
It can be shown in a variety of ways (Taylor expansion comes to mind)
lim n 1 n log ( n ) = 0 ,
and hence
lim n n n = e 0 = 1.

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