Problem $3.(10$ problem$)$ The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its tasks assuming there are $n$ pieces of input $($for example, the number of steps needed to put $n$ numbers in ascending order$).$ To compare the rates of change of two functions, you can find $\underset{n}{lim}\frac{f(n)}{g(n)}.$

Four algorithms for doing the same task have complexities of A: $n^{\frac{3}{2}},B$:$nlo{g}_{2}n,$ C: $n(lo{g}_{2}n{)}^2,$ and $D$:$\sqrt[\phantom{2}]{n}lo{g}_{2}n.$ Rank the algorithms in order of increasing efficiency for large values of $n.$ Graph the complexities as they vary with $n$ and comment on your observations. Show all the work that leads to your answer.