Calculus - limit of a function with logarithms Compute lim_(x -> (0_+)) (ln (x ln a) ln ((ln (ax))/(ln(x/a))))

Noe Cowan

Noe Cowan

Answered question

2022-11-16

Calculus - limit of a function with logarithms
Compute
lim x 0 + ln ( x ln a ) ln ( ln ( a x ) ln ( x a ) )
where a > 1
I am trying to get to a result withou using any advanced methods or even things such as l'Hospital's rule etc..
I got to a phase where I took the limit of the first logarithm which we can see tends to 0 from rewriting it as ln a x . Then I wanted to make some adjustments to the second part of the expression and I got to the stage where I have the limit of
( ln a 2 ) ( x a a )
That wont give me exact result but I should be able to justify that it the expression is defined and by multiplying it with the first limit which is 0, the result should also be 0
Can somebody please tell me how correct or wrong I am? Thanks.

Answer & Explanation

Jackson Trevino

Jackson Trevino

Beginner2022-11-17Added 14 answers

Let b = ln a , y = ln x. Then the limit in question is
L = lim y ( y + ln b ) ln ( y + b y b ) = lim y ( y + ln b ) ln ( y b y + b )
If this limit exists, then
e L = lim y ( y b y + b ) y + ln b = lim y ( 1 2 b y + b ) y + ln b = e 2 b
So, L = 2 b = 2 ln a

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