\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{1+x^2}-\sqrt{1-2x}}{x^2+x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{1+x^2}-1+1-\sqrt{1-2x}}{x\left(x+1\right)}\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{\dfrac{1+x^2-1}{\sqrt[3]{\left(1+x^2\right)^2}+\sqrt[3]{1+x^2}+1}+\dfrac{1-1+2x}{1+\sqrt{1-2x}}}{x\left(x+1\right)}\right)\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{\dfrac{x^2}{\sqrt[3]{\left(x^2+1\right)^2}+\sqrt[3]{x^2+1}+1}+\dfrac{2x}{\sqrt{1-2x}+1}}{x\left(x+1\right)}\right)\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{\dfrac{x}{\sqrt[3]{\left(x^2+1\right)^2}+\sqrt[3]{x^2+1}+1}+\dfrac{2}{\sqrt{1-2x}+1}}{x+1}\right)\)
\(=\left(\dfrac{\dfrac{0}{\sqrt[3]{\left(0^2+1\right)^2}+\sqrt[3]{0^2+1}+1}+\dfrac{2}{\sqrt{1-2\cdot0}+1}}{0+1}\right)\)
\(=\left(\dfrac{2}{1+1}:1\right)=\dfrac{2}{2}=1\)
