I have one question about limits: it is required to find the limit lim n → ∞ 3 n + 2 n n

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I have one question about limits: it is required to find the limit             lim              n        →        ∞                                      3          n                +                  2          n                    n

Isla Klein · Accepted Answer

Notice that  (      3    n    +      2    n        )          1              /            n        =            (              5        n            ⋅                                    3            n                    +                      2            n                                    5          n                    )              1              /            n        =  5  ⋅            [                        (                      3            5                    )                n            +                        (                      2            5                    )                n            ]              1              /            n        .Thus, when you concluded that the second limit is 0 you virtually assumed that      lim          n      →      ∞                  [                        (                      3            5                    )                n            +                        (                      2            5                    )                n            ]              1              /            n        =  0  ,which is false. Your assumption is that since      lim          n      →      ∞                  (              3        5            )        n    +            (              2        5            )        n    =  0  ,that the former limit must also be 0 However, this is not the case: in the cases where the exponent has limit 0 it is not sufficient for the base to have limit 0 in order for the entire power to have limit 0 The exponent is   1      /    n  , and notice that      lim          n      →      ∞            1    n    =  0.As such, your second calculation is simply incorrect. In general, if      lim          n      →      ∞            a    n    =  0and      lim          n      →      ∞            b    n    =  0  ,then      lim          n      →      ∞                          b        n                            a        n            can be equal to any nonnegative real number, or it can even not exist. You cannot simply conclude it to be 0.

babyagelesszj · Accepted Answer

The expression just equals:-      lim          n      →      ∞        3            (      1      +      (              2        3                    )                  n                    )                      1        n            Now this is       1          ∞       form.You can use this whenever you run into this       1          ∞       form      lim          x      →      a        f  (  x      )          g      (      x      )        =  exp  ⁡  (      lim          x      →      a        (  f  (  x  )  −  1  )  g  (  x  )  )when       lim          x      →      a        f  (  x      )          g      (      x      )       is       1          ∞       form.So using this we get it as :-  3  exp  ⁡  (      lim          n      →      ∞                  (              2        3                    )                  n                      n    )  =  3  exp  ⁡  (  0  )  =  3In general whenever you have something like      lim          n      →      ∞                  (              a                  1                          n                    +              a                  2                          n                    +      .      .      .      +              a                  m                          n                    )                      1        n            where       a          i       &#039;s are distinct positive real numbers.You can use the above method and immediately write the answer as             max              1        ≤        i        ≤        m                    a              i            That is we are strongly using the fact that       a          n        →  0 when   0  &amp;lt;  a  &amp;lt;  1

I have one question about limits: it is required to find the limit <mstyle displaystyle="true" sc

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