Question

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

Answer 1

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

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

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

\(=\dfrac{1}{2}+0=\dfrac{1}{2}\)