$\lim_{x \to 1} $ $\frac{2-\sqrt[3]{1+7x}}{x-1}$
$=\lim_{x \to 1} $ $\frac{8-1-7x}{(x-1)[4+2.\sqrt[3]{1+7x}+(\sqrt[3]{1+7x})^2]}$
$=\lim_{x \to 1} $ $\frac{7(1-x)}{(x-1)[4+2.\sqrt[3]{1+7x}+(\sqrt[3]{1+7x})^2]}$
$=\lim_{x \to 1} $ $\frac{-7}{4+2.\sqrt[3]{1+7x}+(\sqrt[3]{1+7x})^2}$
$=\frac{-7}{4+2.\sqrt[3]{1+7}+(\sqrt[3]{1+7})^2}$
$=\frac{-7}{12}$