c. \mathop {\lim }\limits_{x \to 0}\frac{2\sqrt{1+x}-2+2-\sqrt[3]{8-x}}{x}=\mathop {\lim }\limits_{x \to 0}(\frac{2\sqrt{1+x}-2}{x}+\frac{2-\sqrt[3]{8-x}}{x})=\mathop {\lim }\limits_{x \to 0}(\frac{4x}{x(2\sqrt{1+x}+2)}+\frac{x}{x[4+2\sqrt[3]{8-x}+(\sqrt[3]{8-x})^2]})
=\mathop {\lim }\limits_{x \to 0}(\frac{4}{2\sqrt{1+x}+2}+\frac{1}{4+2\sqrt[3]{8-x}+(\sqrt[3]{8-x})^2})
=\mathop {\lim }\limits_{x \to 0}(\frac{4}{2+2}+\frac{1}{4+4+4})=\frac{13}{12}