Let x be a negative real number. I am trying to find the following limit of a sequence: lim n → ∞ ( 1 − 1 − e − x ( e − 1 ) n 1 − e ) n .

Question

Let x be a negative real number. I am trying to find the following limit of a sequence:      lim          n      →      ∞                  (      1      −                        1          −                      e                          −                                                x                  (                  e                  −                  1                  )                                n                                                              1          −          e                    )        n    .

hofyonlines5 · Accepted Answer

I&#039;ll call the expression  f  (  n  )  =            (      1      −                        1          −                      e                          −                              x                (                e                −                1                )                                  /                                n                                                              1          −          e                    )                    n        =            (                        e          −                      e                          −              x              (              e              −              1              )                              /                            n                                                e          −          1                    )                    n      For a given negative real x, we can consider this f a function of real values n with   n  &amp;gt;  −  x  (  e  −  1  )Since the behavior as   n  →  ∞ looks like a &quot;      1    ∞  &quot; case, we can try taking the logarithm and then applying L&#039;Hopital&#039;s rule.                    ln        ⁡        f        (        n        )                            =        ln        ⁡                              (                                          e                −                                  e                                      −                    x                    (                    e                    −                    1                    )                                          /                                        n                                                                              e                −                1                                      )                                            n                                              ln        ⁡        f        (        n        )                            =        n                  [          ln          ⁡                      (            e            −                          e                              −                x                (                e                −                1                )                                  /                                n                                      )                    −          ln          ⁡          (          e          −          1          )          ]                                    ln        ⁡        f        (        n        )                            =                              ln            ⁡                          (              e              −                              e                                  −                  x                  (                  e                  −                  1                  )                                      /                                    n                                            )                        −            ln            ⁡            (            e            −            1            )                                1                          /                        n                                                        lim                      n            →            ∞                          ln        ⁡        f        (        n        )                            =                  lim                      n            →            ∞                                                              1                              e                −                                  e                                      −                    x                    (                    e                    −                    1                    )                                          /                                        n                                                                                      (              −                              e                                  −                  x                  (                  e                  −                  1                  )                                      /                                    n                                            )                                                      x                (                e                −                1                )                                            n                2                                                          −            1                          /                                      n              2                                                                    lim                      n            →            ∞                          ln        ⁡        f        (        n        )                            =                  lim                      n            →            ∞                                                x            (            e            −            1            )                          e                              −                x                (                e                −                1                )                                  /                                n                                                          e            −                          e                              −                x                (                e                −                1                )                                  /                                n                                                                                  lim                      n            →            ∞                          ln        ⁡        f        (        n        )                            =                              x            (            e            −            1            )                          e              0                                            e            −                          e              0                                                                    lim                      n            →            ∞                          ln        ⁡        f        (        n        )                            =        x                            ln        ⁡                  (                      lim                          n              →              ∞                                f          (          n          )          )                                    =        x                                      lim                      n            →            ∞                          f        (        n        )                            =                  e          x

Quintin Stafford · Accepted Answer

Notice that as   n  →  +  ∞ we have      e                            −          x          (          e          −          1          )                n              ∼  1  −            x      (      e      −      1      )        n    +  O  (  1      /        n    2    )hence      lim          n      →      +      ∞                  (      1      −                                    x            (            e            −            1            )                                (            1            −            e            )            n                              )        n    =      lim          n      →      +      ∞                  (      1      +                        x          n                    )        n    =      e    x

Let x be a negative real number. I am trying to find the following limit of a sequence: <munder>

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New Questions in Calculus 1