$\begin{array}{l}1)\quad \lim(\sqrt[3]{n+2} - \sqrt[3]{n})\\ = \lim\dfrac{(\sqrt[3]{n+2} - \sqrt[3]{n})(\sqrt[3]{(n+2)^2} + \sqrt[3]{n(n+2)} + n}{\sqrt[3]{(n+2)^2} + \sqrt[3]{n(n+2)} + \sqrt[3]{n^2}}\\ = \lim\dfrac{2}{\sqrt[3]{(n+2)^2} + \sqrt[3]{n(n+2)} + \sqrt[3]{n^2}}\\ = \dfrac{\dfrac{2}{\sqrt[3]{n^2}}}{\sqrt[3]{\left(1 + \dfrac2n\right)^2} + \sqrt[3]{1 + \dfrac2n} + 1}\\ =\dfrac{0}{\sqrt[3]{(1+0)^2} + \sqrt[3]{1+0} +1}\\ = 0\\ 2)\quad \lim(\sqrt[3]{n^3 - 3n^2} -n)\\ = \lim\dfrac{(\sqrt[3]{n^3 - 3n^2} -n)(\sqrt[3]{(n^3 - 3n^2)^2} + n\sqrt[3]{n^3 - 3n^2} + n^2}{\sqrt[3]{(n^3 - 3n^2)^2} + n\sqrt[3]{n^3 - 3n^2} + n^2}\\ = \lim\dfrac{-3n^2}{\sqrt[3]{(n^3 - 3n^2)^2} + n\sqrt[3]{n^3 - 3n^2} + n^2}\\ = \lim\dfrac{-3}{\sqrt[3]{\left(1 - \dfrac3n\right)^2} + \sqrt[3]{1 - \dfrac3n} + 1}\\ = \dfrac{-3}{\sqrt[3]{(1-0)^2} + \sqrt[3]{1-0} +1}\\ = \dfrac{-3}{3}\\ =-1\\ 3)\quad \lim(\sqrt[3]{n^3 + 3} - \sqrt{n^2+2})\\ = \lim(\sqrt[3]{n^3 + 3} - n + n - \sqrt{n^2+2})\\ = \lim(\sqrt[3]{n^3 + 3} -n) + \lim(n - \sqrt{n^2+2})\\ = \lim\dfrac{(\sqrt[3]{n^3 + 3} -n))(\sqrt[3]{(n^3 + 3)^2} +n\sqrt[3]{n^3 + 3} + n^2}{\sqrt[3]{(n^3 + 3)^2} +n\sqrt[3]{n^3 + 3} + n^2} + \lim\dfrac{(n - \sqrt{n^2+2})(n + \sqrt{n^2+2})}{n + \sqrt{n^2+2}}\\ = \lim\dfrac{3}{\sqrt[3]{(n^3 + 3)^2} +n\sqrt[3]{n^3 + 3} + n^2} + \lim\dfrac{-2}{n + \sqrt{n^2+2}}\\ = \lim\dfrac{\dfrac{3}{n^2}}{\sqrt[3]{\left(1 + \dfrac{3}{n^3}\right)^2} + \sqrt[3]{1 + \dfrac{3}{n^3}} + 1} - \lim\dfrac{\dfrac{2}{n}}{1 + \sqrt{1 + \dfrac{2}{n^2}}}\\ = \dfrac{0}{\sqrt[3]{(1+0)^2} + \sqrt[3]{1+0} +1} - \dfrac{0}{1 + \sqrt{1+0}}\\ = 0\\ 4)\quad \lim(n + 1 - \sqrt{n^2 + n})\\ =\lim\dfrac{(n + 1 - \sqrt{n^2 + n})(n + 1 + \sqrt{n^2 + n})}{n + 1 + \sqrt{n^2 + n}}\\ = \lim\dfrac{n^2 + 2n + 1 - (n^2 + n)}{n + 1 + \sqrt{n^2 + n}}\\ = \lim\dfrac{n +1}{n+1+\sqrt{n^2 +n}}\\ = \lim\dfrac{1 + \dfrac1n}{1 + \dfrac1n + \sqrt{1 + \dfrac1n}}\\ = \dfrac{1+0}{1+0+\sqrt{1+0}}\\ =\dfrac12\\ 5)\quad \lim\dfrac{\sqrt{n^2 + n} - n}{\sqrt{4n^2 +3n} -2n}\\ = \lim\dfrac{(\sqrt{n^2 + n} - n)(\sqrt{n^2 + n} + n)(\sqrt{4n^2 +3n} +2n)}{(\sqrt{n^2 + n} + n)(\sqrt{4n^2 +3n} -2n)(\sqrt{4n^2 +3n} +2n)}\\ =\lim\dfrac{n(\sqrt{4n^2 +3n} +2n)}{(\sqrt{n^2 +n} +n)3n}\\ = \dfrac13\lim\dfrac{\sqrt{4n^2 +3n} +2n}{\sqrt{n^2 +n} +n}\\ = \dfrac13\lim\dfrac{\sqrt{4 + \dfrac3n} + 2}{\sqrt{1 + \dfrac1n} + 1}\\ = \dfrac13\cdot\dfrac{\sqrt{4 +0} +2}{\sqrt{1+0} +1}\\ = \dfrac13\cdot 2\\ =\dfrac23 \end{array}$
