$1)$Use the $lon-N$ definition of convergence to verify the following limits:

$($a$)\underset{n}{lim}\frac{2}{n}=0$

$($b$)\underset{n}{lim}\frac{n+1}{n}=1$

$($c$)\underset{n}{lim}(1-\frac{n}{n+1})=0$

$2)$ Guess the value of $\underset{n}{lim}\frac{3n^2}{n^2+1},$ then use the $lon-N$ definition to verify your guess.

$3)$ Prove that the sequence $x_n=n+(-1{)}^n$ diverges.