\(\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{1+x}-\sqrt{1-x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt[3]{1+x}-1+1-\sqrt{1-x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{x+1-1}{\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{x+1}+1}+\dfrac{1-1+x}{1+\sqrt{1-x}}\right)\cdot\dfrac{1}{x}\)
\(=\lim\limits_{x\rightarrow0}\left(\dfrac{1}{\sqrt[3]{\left(x+1\right)^2}+\sqrt[3]{x+1}+1}+\dfrac{1}{1+\sqrt{1-x}}\right)\)
\(=\left(\dfrac{1}{\sqrt[3]{\left(0+1\right)^2}+\sqrt[3]{0+1}+1}+\dfrac{1}{1+\sqrt{1-0}}\right)\)
\(=\dfrac{1}{3}+\dfrac{1}{2}=\dfrac{5}{6}\)