Evaluate the following limits$.$ Do not use the Fundamental Theorem of Calculus when computing any integral.

$($a$)\underset{n+}{lim}{\displaystyle \underset{k=1}{\overset{n}{}}}\frac{\sqrt[\phantom{2}]{n^2-k^2}}{n^2}$

$($b$)\underset{n+}{lim}\frac{1}{n^2}(\sqrt[\phantom{2}]{2n-1}+\sqrt[\phantom{2}]{4n-4}+\sqrt[\phantom{2}]{6n-9}+$cdots$+\sqrt[\phantom{2}]{(2n{)}^2-(2n{)}^2})$