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