$\begin{array}{l}a)\quad \lim(\sqrt{n^2 +2n} - n - 1)\\ = \lim\dfrac{(\sqrt{n^2 +2n} - n - 1)(\sqrt{n^2 +2n} + n + 1)}{\sqrt{n^2+2n} + n + 1}\\ = \lim\dfrac{n^2 + 2n - (n+1)^2}{\sqrt{n^2 +2n} + n + 1}\\ = \lim\dfrac{-1}{\sqrt{n^2 -2n} + n + 1}\\ = \lim\dfrac{-\dfrac{1}{n}}{\sqrt{1 - \dfrac2n} + 1 + \dfrac1n}\\ = \dfrac{-0}{\sqrt{1-0} + 1 + 0}\\ = 0\\ d)\quad \lim(\sqrt[3]{2n - n^3} +n - 1)\\ = \lim\dfrac{(\sqrt[3]{2n - n^3} +n - 1)(\sqrt[3]{(2n-n^3)^2} - (n-1)\sqrt[3]{2n-n^3} + (n-1)^2}{\sqrt[3]{(2n-n^3)^2} - (n-1)\sqrt[3]{2n-n^3} + (n-1)^2}\\ = \lim\dfrac{2n - n^3 + (n-1)^3}{\sqrt[3]{(2n-n^3)^2} - (n-1)\sqrt[3]{2n-n^3} + (n-1)^2}\\ = \lim\dfrac{-3n^2 + 5n- 1}{\sqrt[3]{(2n-n^3)^2} - (n-1)\sqrt[3]{2n-n^3} + (n-1)^2}\\ = \lim\dfrac{-3 + \dfrac5n - \dfrac{1}{n^2}}{\sqrt[3]{\left(\dfrac{2}{n^2} - 1\right)^2} - \left(1 - \dfrac1n\right)\sqrt[3]{\dfrac{2}{n^2} - 1} + \left(1 - \dfrac1n\right)^2}\\ = \dfrac{-3 +0-0}{\sqrt[3]{(0-1)^2} + 1 + (1-0)^2}\\ =-1 \end{array}$
