Evaluate the limit : lim n → ∞ f ( 3 2 ( 3 + 7 ) n ) g ( 1 2 ( 2 + 2 ) n )

Question

Evaluate the limit :       lim          n      →      ∞                  f              (                  3          2                (        3        +                  7                          )          n                )                    g              (                  1          2                (        2        +                  2                          )          n                )

sepolturamo · Accepted Answer

Let  a  =  (  3  +      7        )    n    ,  b  =  (  3  −      7        )    n    ,  c  =  a  +  band  p  =  (  2  +      2        )    n    ,  q  =  (  2  −      2        )    n    ,  r  =  p  +  qso that   a  ,  b  ,  c  ,  p  ,  q  ,  r are variables depending on nLet us note that   c  ,  r are positive integers (via binomial theorem) and   b  ,  q tend to 0 as   n  →  ∞. The expression under limit can be written as            f      (      3      a              /            2      )              g      (      p              /            2      )        =            f      (      3      c              /            2      −      3      b              /            2      )              g      (      r              /            2      −      q              /            2      )      and by periodic nature of   f  ,  g the above expression equals            f      (      −      3      b              /            2      )              g      (      −      q              /            2      )        =            f      (      −      3      b              /            2      )              /            (      −      3      b              /            2      )              g      (      −      q              /            2      )              /            (      −      q              /            2      )        ⋅            3      b        q  The first fraction tends to   1      /    2 and hence the limit in question is equal to the limit of   3  b      /    2  qNow      b    q    =            (                        3          −                      7                                    2          −                      2                              )        n    →  0and hence desired limit is 0

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