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

ndevunidt 2022-10-14 Answered
Doubts regarding limits and logarithms
Lets 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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Answers (1)

Claire Love
Answered 2022-10-15 Author has 14 answers
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 n
obviously doesn'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't exist.
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