$\lim (\sqrt[3]{n^3 - 3n^2 + 1} - \sqrt{n^2 + 4n})$
$= lim (\sqrt[3]{n^3 - 3n^2 + 1} - n) + lim (n - \sqrt{n^2 + 4n})$
$= lim \dfrac{n^3 - 3n^2 + 1 - n^3}{(\sqrt[3]{n^3 - 3n^2 + 1})^2 + n\sqrt[3]{n^3 - 3n^2 + 1} + n^2} + lim \dfrac{n^2 - n^2 - 4n}{n + \sqrt{n^2 + 4n}}$
$= lim \dfrac{\dfrac{1}{n^2} - 3}{(\sqrt[3]{1 - \dfrac{3}{n} + \dfrac{1}{n^3}})^2 + \sqrt[3]{1 - \dfrac{3}{n} + \dfrac{1}{n^3}} + 1} + lim \dfrac{-4}{1 + \sqrt{1 + \dfrac{4}{n}}}$
`= (0 - 3)/(1 + 1 + 1) + (-4)/(1 + 1)`
`= -1 - 2`
`= -3`
