$\begin{array}{l} x - \sqrt x + 7 = {\left( {\sqrt x - \dfrac{1}{2}} \right)^2} + \dfrac{{27}}{4} > 0\\ A = \dfrac{{\sqrt x + 1}}{{x - \sqrt x + 7}} - \dfrac{1}{3} + \dfrac{1}{3}\\ A = \dfrac{{ - x + \sqrt x - 7 + 3\left( {\sqrt x + 1} \right)}}{{3\left( {x - \sqrt x + 7} \right)}} + \dfrac{1}{3}\\ A = \dfrac{{ - x + 4\sqrt x - 4}}{{3\left( {x - \sqrt x + 7} \right)}} + \dfrac{1}{3}\\ A = \dfrac{{ - {{\left( {\sqrt x - 2} \right)}^2}}}{{3\left( {x - \sqrt x + 7} \right)}} + \dfrac{1}{3} \le \dfrac{1}{3}\\ \Rightarrow \max A = \dfrac{1}{3} \Rightarrow \sqrt x = 2 \Rightarrow x = 4 \end{array}$
