What is the limit of multiple logarithm quotient

kinkonillohup

kinkonillohup

Answered question

2022-03-16

What is the limit of multiple logarithm quotient log2(log2(n))log2(n)
Could somebody check if this is correct?
limnlog2(log2(n))log2(n)
I exponantiate the numerator and the denominator with 2
(log2(log2(n)))2(log2(n))2
=log2(n)n
I extract the constant from the logarithm
=log2(e)limnln(n)n
Using de l'Hospital:
=log2(e)limn1n1
=log2(e)limn1n=0
Is that correct?

Answer & Explanation

Keon Moore

Keon Moore

Beginner2022-03-17Added 2 answers

HINT Let n=22k.Then the limit becomes that of k2k as k
Roland Ramsey

Roland Ramsey

Beginner2022-03-18Added 4 answers

Instead of squaring, it's possible to dive right into L'Hôpital's rule, since as n you get the indeterminate form , as you seem to have noticed. It doesn't make much sense to 'extract the constant from the logarithm,' but instead you should have
limnlog2(log2(n))log2(n)=limnlog2(log2(n))log2(n)=limn1nln2lnn1nln2=limn1lnn.

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