$\lim(\sqrt{n^2+n+1}-\sqrt[3]{n^3+1})$
$=\lim(\sqrt{n^2+n+1}-n+n-\sqrt[3]{n^3+1})$
$=\lim\Big( \dfrac{n+1}{\sqrt{n^2+n+1}+n} -\dfrac{1}{n^2+n.\sqrt[3]{n^3+1}+\sqrt[3]{n^2+1}^2} \Big)$
$=\lim\Big( \dfrac{1+\dfrac{1}{n}}{\sqrt{1+\dfrac{1}{n}+\dfrac{1}{n^2}}+1} - \dfrac{ \dfrac{1}{n^2}}{1+\sqrt[3]{1+\dfrac{1}{n^3}}+ \sqrt[3]{1+\dfrac{1}{n^2}}^2}\Big)$
$=\dfrac{1}{2}-0$
$=\dfrac{1}{2}$
