Đáp án:
\[\lim \left( {\sqrt[3]{{8{n^3} - n}} - 2n} \right) = 0\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\lim \left( {\sqrt[3]{{8{n^3} - n}} - 2n} \right)\\
= \lim \frac{{\left( {\sqrt[3]{{8{n^3} - n}} - 2n} \right)\left( {{{\sqrt[3]{{8{n^3} - n}}}^2} + 2n.\sqrt[3]{{8{n^3} - n}} + 4{n^2}} \right)}}{{{{\sqrt[3]{{8{n^3} - n}}}^2} + 2n.\sqrt[3]{{8{n^3} - n}} + 4{n^2}}}\\
= \lim \frac{{{{\left( {\sqrt[3]{{8{n^3} - n}}} \right)}^3} - {{\left( {2n} \right)}^3}}}{{{{\sqrt[3]{{8{n^3} - n}}}^2} + 2n.\sqrt[3]{{8{n^3} - n}} + 4{n^2}}}\\
= \lim \frac{{ - n}}{{{{\sqrt[3]{{8{n^3} - n}}}^2} + 2n.\sqrt[3]{{8{n^3} - n}} + 4{n^2}}}\\
= \lim \frac{{ - \frac{1}{n}}}{{{{\sqrt[3]{{8 - \frac{1}{{{n^2}}}}}}^2} + 2.\sqrt[3]{{8 - \frac{1}{{{n^2}}}}} + 4}}\\
= \frac{0}{{{{\sqrt[3]{8}}^2} + 2.\sqrt[3]{8} + 4}} = 0
\end{array}\)
