Ordering by asymptotic growth rates $(4$ points$)$: Rank the following functions by order of growth.

This means finding an arrangement g$\_(1),$g$\_(2),$dots,g$\_(9)$ of the functions that satisfy g$\_(1)=\backslash$Omega $($g$\_(2)),$g$\_(2)=$

$\backslash$Omega $($g$\_(3)),$dotsg$\_(7)=\backslash$Omega $($g$\_(8))$

g$\_(1)=$log$($log$($n$))$

g$\_(2)=2^($log$($n$))$

g$\_(3)=2^(\backslash$sqrt$(2\backslash$times log$($n$)))$

g$\_(4)=$n$^(8\mathrm{.}2)+$log$($n$)$

g$\_(5)=$n$^(8\mathrm{.}2)+$n $!$

g$\_(6)=$n$^(2024)+$n$^(512)+$n$\backslash$times log$($n$)$

g$\_(7)=$e$^($n$)+$e$^($ln$($n$))$

g$\_(8)=\backslash$sqrt$($log$($n$))$

Tip: Work in stepwise format. Find the fastest growing function first, and then the next fast one