How to prove that lim n → ∞ n n = 1 ?

Question

How to prove that             lim              n        →        ∞                    n      n        =    1   ?

Dillan Foley · Accepted Answer

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&#039;Hopital&#039;s Rule)      lim          n      →      ∞                  ln      ⁡      (      n      )        n    =      lim          n      →      ∞                  1              /            n        1    =      lim          n      →      ∞            1    n    =  0Therefore, you have      lim          n      →      ∞            n    n    =      e    0    =  1

DinamisGr · Accepted Answer

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.

How to prove that lim_(n->infty) sqrt[n](n)=1 ?

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