\(\begin{array}{l}\text { TXĐ: } D=\mathbb{R} \backslash\{-m\} \\ y^{\prime}=\frac{m^{2}-1}{(x+m)^{2}}\end{array}\)

Hàm số \(y=\frac{m x+1}{x+m}\) đồng biến trên khoảng \((2 ;+\infty)\left\{\begin{array}{l}m^{2}-1\gt 0 \\ -m \notin(2 ;+\infty)\end{array}\right.\)

\(\begin{array}{l}\Leftrightarrow y^{\prime}\gt 0, \forall x \in(2 ;+\infty) \Leftrightarrow\left\{\begin{array}{l}m^{2}-1>0 \\ -m \leq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}m \in(-\infty ;-1) \cup(1 ;+\infty) \\ -m \leq 2\end{array} \Leftrightarrow\left\{\begin{array}{l}m \in(-\infty ;-1) \cup(1 ;+\infty) \\ m \geq-2\end{array}\right.\right.\right. \\ \Leftrightarrow m \in[-2 ;-1) \cup(1 ;+\infty) .\end{array}\)