Chọn A
Xét \(x\in \left(1;e\right)\Rightarrow \ln x\in \left(0;1\right). \)
Ta có:
\(y'=\frac{\left(\ln x-6\right)^{{'} } \left(\ln x-2m\right)-\left(\ln x-2m\right)^{{'} } \left(\ln x-6\right)}{\left(\ln x-2m\right)^{2} } =\frac{-2m+6}{\left(\ln x-2m\right)^{2} } .\frac{1}{x} \)
Hàm số đồng biến trên khoảng
\(\left(1;e\right)\Leftrightarrow y'>0,\forall x\in \left(1;e\right)\Leftrightarrow \left\{\begin{array}{l} {-2m+6>0} \\ {2m\rlap{/}\in \left(0;1\right)} \end{array}\right.\)
\( \Leftrightarrow \left\{\begin{array}{l} {m<3} \\ {\left[\begin{array}{l} {m\le 0} \\ {m\ge \frac{1}{2} } \end{array}\right. } \end{array}\right. \Leftrightarrow \left[\begin{array}{l} {m\le 0} \\ {\frac{1}{2} \le m<3} \end{array}\right. .\)
Vậy \(S=\left\{0;1;2\right\}. \)