Asymptotics of logarithms of functions If I know that <munder> <mo form="prefix">lim

Daphne Fry

Daphne Fry

Answered question

2022-05-14

Asymptotics of logarithms of functions
If I know that lim x f ( x ) g ( x ) = 1, does it follow that lim x log f ( x ) log g ( x ) = 1 as well? I see that this definitely doesn't hold for e f ( x ) e g ( x ) (take f ( x ) = x + 1 and g ( x ) = x), but I'm not sure how to handle the other direction.

Answer & Explanation

Kylan Simon

Kylan Simon

Beginner2022-05-15Added 17 answers

No, the asymptotic equality of the logarithms doesn't follow. Consider
f ( x ) = 1 + 1 x ; g ( x ) = 1 + 1 x 2 .
Ignoring lower-order terms, we have
log f ( x ) log g ( x ) = 1 / x + O ( x 2 ) 1 / x 2 + O ( x 4 ) 1 / x 1 / x 2 = x .
However, from f g we have
lim x log f ( x ) log g ( x ) = 0 ,
so if f and g stay away from 1, you have an even stronger result than mere asymptotic equality.
Jayden Mckay

Jayden Mckay

Beginner2022-05-16Added 3 answers

It does not follow. Take the example of f ( x ) = e x + 1 and g ( x ) = 1. Then
lim x f ( x ) g ( x ) = lim x e x + 1 1 = 1
However,
lim x log f ( x ) log g ( x ) = lim x log ( e x + 1 ) log 1
Does not exist.

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