Đặt 

A

- Hướng dẫn giải

Đáp án A

Ta có: $I=\underset{x\to 0}{\lim }\frac{x{\log }_a\left(1-2x\right)+1-\cos x}{x^2},0<a\ne 1$ Chú ý: $\underset{x\to 0}{\lim }\frac{\ln (1+ax)}{ax}=1;\underset{x\to 0}{\lim }\frac{\sin ax}{ax}=1.$

$\Rightarrow I=\underset{x\to 0}{\lim }\frac{{\log }_a\left(1-2x\right)}{x}+\underset{x\to 0}{\lim }\frac{2{\sin }^2{\displaystyle \frac{x}{2}}}{4{\left({\displaystyle \frac{x}{2}}\right)}^2}=\underset{x\to 0}{\lim }\frac{-2{\log }_{a}e.\ln \left(1-2x\right)}{-2x}+\frac{1}{2}=\frac{1}{2}-\frac{2}{\ln a}.$