6.
$\lim\dfrac{\sqrt[3]{2n^2-n^3}+n}{\sqrt{n^2+n}-n}$
$=\lim\dfrac{ (\sqrt[3]{2n^2-n^3}+n)(\sqrt{n^2+n}+n)}{n}$
$=\lim\dfrac{2n^2(\sqrt{n^2+n})}{n.(\sqrt[3]{2n^2-n^3}^2-n\sqrt[3]{2n^2-n^3}+n^2}$
$=\lim\dfrac{ 2n.\sqrt{n^2+n}}{\sqrt[3]{2n^2-n^3}^2-n.\sqrt[3]{2n^2-n^3}+n^2}$
$=\lim\dfrac{ 2n^2\sqrt{1+\dfrac{1}{n}} }{n^2\sqrt[3]{\dfrac{2}{n}-1}^2-n^2\sqrt[3]{\dfrac{2}{n}-1}+n^2}$
$=\lim\dfrac{ 2\sqrt{1+\dfrac{1}{n}} }{\sqrt[3]{\dfrac{2}{n}-1}^2-\sqrt[3]{\dfrac{2}{n}-1}+1}$
$=\dfrac{2.1}{1-1+1}=2$
7.
$\lim(\sqrt{4n^2+2n+3}-2n)$
$=\lim\dfrac{ 2n+3}{\sqrt{4n^2+2n+3}+2n}$
$=\lim\dfrac{2+\dfrac{3}{n}}{\sqrt{4+\dfrac{2}{n}+\dfrac{3}{n^2}}+2}$
$=\dfrac{2}{2+2}$
$=\dfrac{1}{2}$
