$7)$Verify the following limits using the $lon-N$ definition:

$($a$)\underset{n}{lim}\frac{1}{\sqrt[\phantom{2}]{n}}=0$

$($b$)\underset{n}{lim}\frac{n-1}{n^2}=0$

$($c$)\underset{n}{lim}\frac{(n+1{)}^2}{n^2}=1$

$8)$ Prove that the sequence

$x_n=\{\begin{array}{ccc}1,& nis\text{p}\text{r}\text{i}\text{m}\text{e}& \\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

does not converge.