how to tell which function asymptotically grows faster than other?

Question

I read this answer

But functions in my case seems to be complicated to me..

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Which of the following functions asymptotically grows the fastest as $n$ goes to infinity?

$(\log \log(n))!$

$(\log\log(n))^{\log(n)}$

$(\log\log(n))^{\log\log\log(n)}$

$(\log(n))^{\log\log(n)}$

$2^{\sqrt{\log\log(n)}}$

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for example take the first and second function that are

$f(n) = (\log\log(n))!$

$g(n) = (\log\log(n))^{\log(n)}$

now calculating

$$\lim_{n \to \infty} \frac{f(n)}{g(n)}$$

$f(n)$ it's derivative can't be calculated.

Then how can I tell which function asymptotically grows fastest ?

The answer given in the book is $(\log\log(n))^{\log(n)}$

Answer 1

Take logarithms. To compare the second with the first, for example, ee have $$ \log((\log\log n)^{\log n}) = \log n \cdot\log\log\log n \tag{1}$$ On the other hand, we know that $\log(x!) \sim x\log x,$ so that $$ \log(\log \log n)!) \sim \log\log n \cdot \log\log\log n\tag{2} $$ Comparing $(1)$ and $(2)$, we see that $(1)$ grows much faster.

It looks to me like the fourth one is the biggest, just doing it in my head.