\(\lim\limits_{x\rightarrow0}\dfrac{2\sqrt{1+x}-\sqrt[3]{8-x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt{4+4x}-\sqrt[3]{8-x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\sqrt{4+4x}-2+2-\sqrt[3]{8-x}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{4x+4-4}{\sqrt{4x+4}+2}+\dfrac{8-8+x}{4+2\cdot\sqrt[3]{8-x}+\sqrt[3]{\left(8-x\right)^2}}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{\dfrac{4x}{\sqrt{4x+4}+2}+\dfrac{x}{4+2\cdot\sqrt[3]{8-x}+\sqrt[3]{\left(8-x\right)^2}}}{x}\)
\(=\lim\limits_{x\rightarrow0}\dfrac{4}{\sqrt{4x+4}+2}+\dfrac{1}{4+2\cdot\sqrt[3]{8-x}+\sqrt[3]{\left(8-x\right)^2}}\)
\(=\dfrac{4}{\sqrt{4\cdot0+4}+2}+\dfrac{1}{4+2\cdot\sqrt[3]{8-0}+\sqrt[3]{\left(8-0\right)^2}}\)
\(=\dfrac{4}{2+2}+\dfrac{1}{4+2\cdot2+4}\)
\(=1+\dfrac{1}{12}=\dfrac{13}{12}\)
