Compute $$\lim_{x \to {0_+}} {\ln (x \ln a) \ln \left({{\ln (ax)}\over{\ln({x\over a})}}\right)}$$ 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 $$(\ln a^2) \left({x-a\over a}\right)$$ 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.