1) $\lim\dfrac{(2n+1)(n^2 +2)}{1-3n+n^3}$
$=\lim\dfrac{\left(2+\dfrac1n\right)\left(1+\dfrac{2}{n^2}\right)}{\dfrac{1}{n^3} - \dfrac{3}{n^2} + 1}$
$=\dfrac{(2+0)(1+0)}{0-0+1}$
$=2$
2) $\lim(\sqrt{8n^2 + 3n + 2} - 2\sqrt2n + 5)$
$=\lim\dfrac{(\sqrt{8n^2 + 3n + 2} - 2\sqrt2n + 5)(\sqrt{8n^2 + 3n + 2} +2\sqrt2n- 5)}{\sqrt{8n^2 + 3n + 2} + 2\sqrt2n -5}$
$=\lim\dfrac{8n^2 + 3n +2 - (2\sqrt2n-5)^2}{\sqrt{8n^2 + 3n + 2} +2\sqrt2n - 5}$
$= \lim\dfrac{(20\sqrt2 +3)n - 23}{\sqrt{8n^2 + 3n + 2} +2\sqrt2n - 5}$
$=\lim\dfrac{20\sqrt2 + 3 - \dfrac{23}{n}}{\sqrt{8 + \dfrac3n +\dfrac{2}{n^2}} + 2\sqrt2 -\dfrac5n}$
$=\dfrac{20\sqrt2 + 3 - 0}{\sqrt{8 + 0 + 0} + 2\sqrt2 - 0}$
$=\dfrac{20\sqrt2 +3}{4\sqrt2}$
3) $\lim\dfrac{(n+1)^2\sqrt{1+4+7+\dots + (3n+1)}}{(1-2n)^2}$
$=\lim\dfrac{(n+1)^2}{(1-2n)^2}\cdot\lim\sqrt{1+4+7+\dots+(3n-2)+3}$
$=\lim\left(\dfrac{1+\dfrac1n}{\dfrac1n -2}\right)^2\cdot\lim\sqrt{\dfrac{n(3n-1)}{2} +3}$
$=\left(\dfrac{1+0}{0-2}\right)^2\cdot \lim \left(n\sqrt{\dfrac{3 -\dfrac1n}{2} +\dfrac{3}{n^2}}\right)$
$=\dfrac14\cdot +\infty\cdot \sqrt{\dfrac{3 -0}{2} +0}$
$= +\infty$
