\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt[]{1+2x}\left(\sqrt[3]{3x+1}-\left(x+1\right)\right)+\left(x+1\right)\left(\sqrt[]{1+2x}-\left(x+1\right)\right)+x^2+\left(2x+1-\sqrt[]{4x+1}\right)}{1+x-\sqrt[]{2x+1}}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt[]{1+2x}.\dfrac{x^2\left(-x-3\right)}{\sqrt[3]{\left(3x+1\right)^2}+\sqrt[3]{3x+1}+1}+\dfrac{x^2.\left(x+1\right)}{\sqrt[]{1+2x}+x+1}+x^2+\dfrac{x^2}{2x+1+\sqrt[]{4x+1}}}{\dfrac{x^2}{1+x+\sqrt[]{2x+1}}}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{\left(-x-3\right)\sqrt[]{1+2x}}{\sqrt[3]{\left(3x+1\right)^2}+\sqrt[3]{3x+1}+1}+\dfrac{x+1}{\sqrt[]{1+2x}+x+1}+1+\dfrac{1}{2x+1+\sqrt[]{4x+1}}}{\dfrac{1}{1+x+\sqrt[]{2x+1}}}\)
\(=3\)
