But I see you say both are correct, but it seems strange that you can justify a limit by order of infinity

For example

$\displaystyle lim_{x\to\infty}\frac{\ln(\ln(x))}{\ln(x)}=0$

How can we say this is true by the fact that $\displaystyle \ln(\ln(x))\prec\ln(x)$ when this is the defintion of orders of infinity?

It would be like showing

$\displaystyle \frac{d}{dx}\bigg[x^n\bigg]=nx^{n-1}$

But showing it by the difference quotient, and then using L'hopital's, but we cannot do this for we would use the conclusion of our problem in the problem before proving it

Any clarification would be greatly appreciated