Determine the value of a so that the lim x → ∞ ( a 1 x + 1 x ) x = 3

Question

Determine the value of a so that the      lim          x      →      ∞                  (              a                              1            x                              +              1        x            )        x    =  3

Jovan Wong · Accepted Answer

We know that       a          1              /            x       is well defined for   a  &amp;gt;  0. Let   f  (  x  )  =      a          1              /            x        +      1    x    −  1, let us also note that             lim              x        →        ∞              f    (    x    )    =    0  , then      lim          x      →      ∞                  (              a                              1            x                              +              1        x            )        x    =      lim          x      →      ∞                  (      1      +      f      (      x      )      )        x    =      lim          x      →      ∞                  [      (      1      +      f      (      x      )              )                              1                          f              (              x              )                                          ]              x      ⋅      f      (      x      )        =      e                            lim                      x            →            ∞                          x        f        (        x        )            We then focus our attention on                                          lim                          x              →              ∞                                x          f          (          x          )                                    =                  lim                      x            →            ∞                          1        +        x        (                  a                      1                          /                        x                          −        1        )                                          =        1        +                  lim                      x            →            ∞                                                              a                              1                                  /                                x                                      −            1                                1                          /                        x                                                            =        1        +                  lim                      x            →            ∞                                                              −                              a                                  1                                      /                                    x                                            ln              ⁡              a                                      x              2                                            −                          1                              x                2                                                                                      =        1        +                  lim                      x            →            ∞                                    a                      1                          /                        x                          ln        ⁡        a                                          =        1        +        ln        ⁡        a            Therefore  3  =      lim          x      →      ∞                  (              a                              1            x                              +              1        x            )        x    =      e                            lim                      x            →            ∞                          x        f        (        x        )              =      e          1      +      ln      ⁡      a      then we have   a  =  3      /    e

Determine the value of a so that the <munder> <mo movablelimits="true" form="prefix">lim

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