The value of $\underset{\mathrm{x}\to 0}{\lim }\left(\frac{\mathrm{x}}{\sqrt[8]{1-\mathrm{sinx}} - \sqrt[8]{1 + \mathrm{sinx}}}\right)$ is equal to: 

$$\begin{gathered} \underset{\mathrm{x}\to 0}{\lim }\left(\frac{\mathrm{x}}{\sqrt[8]{1-\mathrm{sinx}} - \sqrt[8]{1 + \mathrm{sinx}}}\right) \left( \frac{0}{0} \mathrm{form} \mathrm{L}\text{'} \mathrm{Hospital} \mathrm{Rule}\right) \\ =\underset{\mathrm{x}\to 0}{\lim } \frac{1}{{\displaystyle \frac{1}{8}}{(1-\sin \mathrm{x})}^{-7/8} (-\cos \mathrm{x})-{\displaystyle \frac{1}{8}}{(1+\sin \mathrm{x})}^{-7/8} \cos \mathrm{x}} \\ =\frac{1}{-{\displaystyle \frac{1}{8}}-{\displaystyle \frac{1}{8}}}=\frac{1}{{\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}=-4 \end{gathered}$$