Evaluate lim x → ∞ x 2 ( 4 1 x − 4 1 1 + x )

Question

Evaluate      lim          x      →      ∞            x    2    (      4                  1        x              −      4                  1                  1          +          x                      )

billyfcash5n · Accepted Answer

Write                    4        t            −              4                  t                      /                    (          t          +          1          )                            t      2        =      4          t              /            (      t      +      1      )        ⋅                    4                              t            2                                /                    (          t          +          1          )                    −      1                      t        2                    t        +        1              ⋅      1          t      +      1      and use the standard limit       lim          u      →      0                          a        u            −      1        u    =  log  ⁡  a to find                              lim                      t            →            0                                                              4              t                        −                          4                              t                                  /                                (                t                +                1                )                                                          t            2                                              =                  lim                      t            →            0                                    4                      t                          /                        (            t            +            1            )                          ⋅                  lim                      t            →            0                                                              4                                                t                  2                                                  /                                (                t                +                1                )                                      −            1                                              t              2                                      t              +              1                                      ⋅                  lim                      t            →            0                                    1                      t            +            1                                                            =        1        ⋅        log        ⁡        4        ⋅        1        =        log        ⁡        4

2nalfq8 · Accepted Answer

Mean Value Theorem, for   f  (  x  )  =      4    x  , with       f    ′    (  x  )  =      4    x    ln  ⁡  4: There exists a       ξ    x    ∈            (        1      /    x  ,  1      /    (  x  +  1  )            )      , such that      4                  1        x              −      4                  1                  x          +          1                      =      (          1      x        −          1              x        +        1              )        4                  ξ        x              ln  ⁡  4  =                    4                              ξ            x                              ln      ⁡      4              x      (      x      +      1      )      Clearly       lim          x      →      ∞            ξ    x    =  0, and hence       lim          x      →      ∞            4                  ξ        x              =  1. Hence      lim          x      →      ∞            x    2              (            4                  1        x              −      4                  1                  x          +          1                                )        =      lim          x      →      ∞                  x      2              x      (      x      +      1      )            4                  ξ        x              ln  ⁡  4  =  ln  ⁡  4.

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