Determine whether the sequence converges or diverges, and if it converges, find the limit$.$

$($a$)\{\sqrt[\phantom{2}]{2n^2+2n}-\sqrt[\phantom{2}]{2n^2+3}\}.$

$($b$)\{(\frac{n+2}{n+1}{)}^{\frac{n}{3}}\}$

$($c$)\{(\frac{1-n^3}{3n^2-7n^3})si{n}^{-1}(\frac{\text{I}}{1-5n^2-2n^3})\}$