How to compute lim_(x -> oo)(ln(1+e^(alpha x)))/(ln(1+e^(beta x))), where alpha>0 and beta>0

Elliana Molina

Elliana Molina

Answered question

2022-11-16

How to compute lim x ln ( 1 + e α x ) ln ( 1 + e β x ) ?
Where α > 0 and β > 0

Answer & Explanation

Pignatpmv

Pignatpmv

Beginner2022-11-17Added 22 answers

Use the log functional equation.
log ( 1 + e α x ) = log ( e α x ) + log ( 1 + e α x )
so that
lim x ln ( 1 + e α x ) ln ( 1 + e β x ) = lim x α x + ln ( 1 + e α x ) β x + ln ( 1 + e β x )
The log terms now go to 0 since α , β > 0, and we are left with α β as our limit.
inurbandojoa

inurbandojoa

Beginner2022-11-18Added 11 answers

Write the function as
f ( x ) = ln ( 1 + e α x ) ln ( 1 + e β x ) = ln ( e α x ( 1 + e α x ) ) ln ( e β x ( 1 + e β x ) ) = α x + ln ( 1 + e α x ) β x + ln ( 1 + e β x )
Since, α , β , x are all non-negative, we have
α x β x + ln ( 2 ) f ( x ) α x + ln ( 2 ) β x
Now you should be able to compute the desired limit.

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