Question

Tính lim(\(\sqrt{4n^2+2}.\sqrt[3]{n^3+1}-2n\sqrt[3]{n^3+2}\))

Answer 1

\(lim\left(\sqrt{4n^2+2}\sqrt[3]{n^3+1}-2n\sqrt[3]{n^3+2}\right)\\ =lim\left[\sqrt[3]{n^3+1}\left(\sqrt{4n^2+2}-2n\right)-2n\left(\sqrt[3]{n^3+2}-\sqrt[3]{n^3+1}\right)\right]\\ =lim\left[\dfrac{2\sqrt[3]{n^3+1}}{\sqrt{4n^2+2}+2n}-\dfrac{2n}{\left(\sqrt[3]{n^3+2}\right)^2+\left(\sqrt[3]{n^3+1}\right)^2+\sqrt[3]{n^3+2}\sqrt[3]{n^3+1}}\right]\\ =lim\left[\dfrac{2\sqrt[3]{1+\dfrac{1}{n^3}}}{\sqrt{4+\dfrac{2}{n^2}}+2}-\dfrac{\dfrac{2}{n}}{\left(\sqrt[3]{1+\dfrac{2}{n}}\right)^2+\left(\sqrt[3]{1+\dfrac{1}{n}}\right)^2+\sqrt[3]{1+\dfrac{2}{n}}\sqrt[3]{1+\dfrac{1}{n}}}\right]\)

\(=\dfrac{1}{2}\)