Question

\(\lim\limits_{x\rightarrow0}\dfrac{2x}{1+\sqrt[3]{2x-1}}\)

Answer 1

\(\lim\limits_{x\rightarrow0}\dfrac{2x}{1+\sqrt[3]{2x-1}}\)

\(=\lim\limits_{x\rightarrow0}\dfrac{2x}{\dfrac{1+2x-1}{1-\sqrt[3]{2x-1}+\sqrt[3]{\left(2x-1\right)^2}}}\)

\(=\lim\limits_{x\rightarrow0}\left(2x\cdot\dfrac{1-\sqrt[3]{2x-1}+\sqrt[3]{\left(2x-1\right)^2}}{2x}\right)\)

\(=\lim\limits_{x\rightarrow0}\left(1-\sqrt[3]{2x-1}+\sqrt[3]{\left(2x-1\right)^2}\right)\)

\(=1-\sqrt[3]{2\cdot0-1}+\sqrt[3]{\left(2\cdot0-1\right)^2}\)

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

\(=1-\left(-1\right)+\left(-1\right)=1\)