Asymptotic growth. $[15$ points$]$

For each of the following pairs of functions $f(n)$ and $g(n),$ indicate whether $f(n)$ is in $O,o,,,$ and$/$or $$ of $g(n).($You should either write true or false for each of these five possibilities, or give an exhaustive list of the relationships that hold.$)$ Explanations are not required, but can be provided for possible partial credit.

$($a$)f(n)=2^{2024n},g(n)=n^n$

$($b$)f(n)=n(logn+loglogn),g(n)=(n+loglogn)logn$

$($c$)f(n)=\sqrt[\phantom{2}]{n},g(n)=n^{cos(\sqrt[\phantom{2}]{n})}$

$($d$)f(n)=\frac{\sqrt[\phantom{2}]{n}+n^2}{2024},g(n)=2024n$

$($e$)f(n)={}^n,g(n)=3^n$