$2+h$ آهنگ متوسط تغییر تابع از 2 تا $=\frac{f\left( 2+h \right)-f\left( 2 \right)}{2+h-2}=\frac{\left( 2+h+\frac{1}{2+h} \right)-\left( 2+\frac{1}{2} \right)}{h}=\frac{8}{9}$
$\xrightarrow{h\ne 0}2+h+\frac{1}{2+h}-\frac{5}{2}=\frac{8}{9}h\Rightarrow h+\frac{1}{2+h}-\frac{1}{2}=\frac{8}{9}h$
$\Rightarrow 9h+\frac{9}{2+h}-\frac{9}{2}=8h\Rightarrow h+\frac{9}{2+h}=\frac{9}{2}$
$\Rightarrow \frac{{{h}^{2}}+2h+9}{2+h}=\frac{9}{2}\Rightarrow 2{{h}^{2}}-5h=0\Rightarrow \left\{ \begin{matrix} h=2/5 \\ h=0 \\ \end{matrix} \right.$