$3]$ Rank the following functions by order of growth; that is$,$ find an arrangement$81,82....,817$of the functions below satisfying$810(82),82$ O$(83),....$ go $0(817).$Partition your list into equivalence classes such that f$($n$)$ and g$($z$)$ are in the same class if and only if f$($n$)(8($n$)).$ Robust mathematical proofs are preferred, but some credit might be obtained for correct answers obtained through "hand waving" arguments $($the wilder the waving, the lower the grade$).($And note that "Ign" is "Jog$2$ n$".)$It may help to observe that if f$($n$)$ o$($g$($n$)),$ then f$($n$)$ $ $0($g$($n$)).($i$)($li$)($lii$)16$kn$($v$)1$g$((2$n$)!)($v$)$ n$($vi$)$ n$!($vii$)218720-1($viji$)$ Ign $($ix$)$ Ig$($lgn$)($x$)$v$21$gn$($xi$)($n $+1)!($xii$)($xiii$)300$n $+2\mathrm{.}99793\backslash$times $108($xiv$)($Ign$)$en$($x$)(3/2)*($xvi$)1$gnS$($xvii$)$ n$2"$