Doubts regarding limits and logarithmsLets say you are given this limit lim n → ∞ ( log ⁡ ( n + n n + n 1 / n )That expression is equal to log ⁡ ( lim n → ∞ [ n + n n + n 1 / n ] )isn't it?My question is if I could descompose the limit like this without changing the limit like this log ⁡ ( lim n → ∞ n + lim n → ∞ n n + lim n → ∞ n 1 / n )Could I?

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Doubts regarding limits and logarithmsLets say you are given this limit      lim          n      →      ∞        (  log  ⁡  (  n  +      n    n    +      n          1              /            n        )That expression is equal to  log  ⁡  (      lim          n      →      ∞        [  n  +      n    n    +      n          1              /            n        ]  )isn&#039;t it?My question is if I could descompose the limit like this without changing the limit like this  log  ⁡  (      lim          n      →      ∞        n  +      lim          n      →      ∞            n    n    +      lim          n      →      ∞            n          1              /            n        )Could I?

Claire Love · Accepted Answer

First of all, a sufficient condition to interchange the limit and a function is that the function is continuous. So for   f  (  x  ) continuous we have      lim          n      →      ∞        f  (      a    n    )  =  f  (      lim          n      →      ∞            a    n    )since the logarithm is continuous, you can      lim          n      →      ∞        log  ⁡  (      a    n    )  =  log  ⁡  (      lim          n      →      ∞            a    n    )interchange.To answer the second part of your question: No in this particular case you cannot split the limit! For this to hold all limits need to exist on their own, for example      lim          n      →      ∞        (      a    n    +      b    n    )  =      lim          n      →      ∞        (      1    n    +            3      n      +      1              7      n      −      2        )  =      lim          n      →      ∞            1    n    +      lim          n      →      ∞                  3      n      +      1              7      n      −      2        =  0  +      3    7    =      3    7  works, since both limits       lim          n      →      ∞            a    n    ,      lim          n      →      ∞            b    n   do exist. But for example      lim          n      →      ∞        (  0  )  =      lim          n      →      ∞        (  n  −  n  )  ≠      lim          n      →      ∞        n  −      lim          n      →      ∞        nobviously doesn&#039;t work since the limit of       lim          n      →      ∞        n does not exist. In your case      lim          n      →      ∞        (  log  ⁡  (  n  +      n    n    +      n          1              /            n        )  ≠  log  ⁡  (      lim          n      →      ∞        n  +      lim          n      →      ∞            n    n    +      lim          n      →      ∞            n          1              /            n        )because       lim          n      →      ∞            n    n   and       lim          n      →      ∞        n don&#039;t exist.

Doubts regarding limits and logarithms Lets say you are given this limit lim_(n->oo) ( log(n + n^n + n^(1/n) )

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