Determining lim n → ∞ ( n 1 n − 1 ) n with only elementary mathI am trying to find this limit: lim n → ∞ ( n 1 n − 1 ) n ,I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you

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Determining       lim          n      →      ∞                  (              n                                            1              n                                          −      1      )        n   with only elementary mathI am trying to find this limit:      lim          n      →      ∞                  (              n                                            1              n                                          −      1      )        n    ,I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you

eri1ti0m · Accepted Answer

For   n  ≥  5 we have, using the binomial theorem            (      1      +              1        2            )        n    ≥  1  +      n    2    +            n      (      n      −      1      )        8    ≥  1  +      n    2    +            n      (      5      −      1      )        8    =  1  +  n  &amp;gt;  nThus   1  +      1    2    ≥      n          1              /            n      , or  0  ≤  (      n          1              /            n        −  1      )    n    ≤      1          2      n      (the first inequality being immediate.) Tking the limit as   n tend to   ∞, we get      lim          n      →      ∞        (      n          1              /            n        −  1      )    n    =  0.

anudoneddbv · Accepted Answer

Perhaps the simplest way is to go step by step. First, show that       n          1              /            n        →  1 as   n  →  ∞. You can then conclude that   0  &amp;lt;      n          1              /            n        −  1  &amp;lt;      1    2   for large enough values of   n, meaning that, when   n  →  ∞  , the limit must be   0 (sandwich principle).

Determining lim_(n->oo}(n^((1)/(n))-1)^n with only elementary math I tried using exponential function, but I see no way at the moment. I am not allowed to use any kind of differentiation or other topics of advanced math, only induction and school math are possible. Thank you

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