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