1: \(\lim_{}\left(\sqrt[3]{8n^3+4n^2+1}-\sqrt[3]{8n^3-2}\right)\)
\(=\lim_{}\frac{8n^3+4n^2+1-8n^3+2}{\sqrt[3]{\left(8n^3+4n^2+1\right)^2}+\sqrt[3]{\left(8n^3+4n^2+1\right)\left(8n^3-2\right)}+\sqrt[3]{\left(8n^3-2\right)^2}}\)
\(=\lim_{}\frac{4n^2+3}{\sqrt[3]{\left\lbrack n^3\left(8+\frac{4}{n}+\frac{1}{n^3}\right)^{}\right\rbrack^2}+\sqrt[3]{\left\lbrack n^3\left(8+\frac{4}{n}+\frac{1}{n^3}\right)\cdot n^3\cdot\left(8-\frac{2}{n^3}\right)\right\rbrack}+\sqrt[3]{\left\lbrack n^3\left(8-\frac{2}{n^3}\right)^2\right\rbrack}}\)
\(=\lim_{}\frac{4n^2+3}{n^2\cdot\left(\sqrt[3]{\left(8+\frac{4}{n}+\frac{1}{n^3}\right)^2}+\sqrt[3]{\left(8+\frac{4}{n}+\frac{1}{n^3}\right)\cdot\left(8-\frac{2}{n^3}\right)}+\sqrt[3]{\left(8-\frac{2}{n^3}\right)^2}\right)}\)
\(=\lim_{}\frac{4+\frac{3}{n^2}}{\left(\sqrt[3]{\left(8+\frac{4}{n}+\frac{1}{n^3}\right)^2}+\sqrt[3]{\left(8+\frac{4}{n}+\frac{1}{n^3}\right)\cdot\left(8-\frac{2}{n^3}\right)}+\sqrt[3]{\left(8-\frac{2}{n^3}\right)^2}\right)}\)
\(=\frac{4+0}{\sqrt[3]{\left(8+0+0\right)^2}+\sqrt[3]{\left(8+0+0\right)\left(8-0\right)}+\sqrt[3]{\left(8-0\right)^2}}\)
\(=\frac{4}{4+4+4}=\frac{4}{12}=\frac13\)
2: \(\lim_{}\left(\sqrt[3]{n^3+n^2+1}+\sqrt[3]{8-n^3}\right)\)
\(=\lim_{}\frac{n^3+n^2+1+8-n^3}{\sqrt[3]{\left(n^3+n^2+1\right)^2}-\sqrt[3]{\left(n^3+n^2+1\right)\left(8-n^3\right)}+\sqrt[3]{\left(8-n^3\right)^2}}\)
\(=\lim_{}\frac{n^2+9}{n^2\cdot\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^2}-n^2\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)\left(\frac{8}{n^3}-1\right)}+n^2\cdot\sqrt[3]{\left(\frac{8}{n^3}-1\right)^2}}\)
\(=\lim_{}\frac{1+\frac{9}{n^2}}{\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)^2}-\sqrt[3]{\left(1+\frac{1}{n}+\frac{1}{n^3}\right)\left(\frac{8}{n^3}-1\right)}+\sqrt[3]{\left(\frac{8}{n^3}-1\right)^2}}\)
\(=\frac{1+0}{\sqrt[3]{\left(1+0+0\right)^2}-\sqrt[3]{\left(1+0+0\right)\left(0-1\right)}+\sqrt[3]{\left(0-1\right)^2}}\)
\(=\frac{1}{1-1+1}=\frac11\) =1
3: \(\lim_{}\left(\sqrt[3]{n^3+n^2+2}-n\right)\)
\(=\lim_{}\frac{n^3+n^2+2-n^3}{\sqrt[3]{\left(n^3+n^2+2\right)^2}+n\cdot\sqrt[3]{n^3+n^2+2}+n^2}\)
\(=\lim_{}\frac{n^2+2}{n^2\cdot\sqrt[3]{\left(1+\frac{1}{n}+\frac{2}{n^3}\right)^2}+n^2\cdot\sqrt[3]{1+\frac{1}{n}+\frac{2}{n^3}}+n^2}\)
\(=\lim_{}\frac{1+\frac{2}{n^2}}{\sqrt[3]{\left(1+\frac{1}{n}+\frac{2}{n^3}\right)^2}+\sqrt[3]{1+\frac{1}{n}+\frac{2}{n^3}}+1}=\frac{1+0}{\sqrt1+1+1}=\frac13\)
