What is the value of this formula?

If we only use $b = a$ , it doesn't work: $(a-\frac{a^2}{2a}) \div \color{#D61F06}{\frac{0}{0}}$ So we can only simplify. $\newline (\sqrt{ab}-\frac{ab}{a+\sqrt{ab}})\div \frac{\sqrt{ab}-b}{a-b}\\ =\frac{a\sqrt{ab}+ab-ab}{a+\sqrt{ab}} \times \frac{a-b}{\sqrt{ab}-b} \\ =\frac{(a-b)\sqrt{ab} \times a}{(a+\sqrt{ab})(\sqrt{ab}-b)} \\ =\frac{a(a-b)\sqrt{ab}}{a\sqrt{ab}-b\sqrt{ab}-ab+ab} \\ =\frac{a(a-b)\sqrt{ab}}{(a-b)\sqrt{ab}}\\ =\boxed{a}$

$\lim_{b\rightarrow{a}}\sqrt{ab}(1-\dfrac{\sqrt{ab}}{a+\sqrt{ab}})× \dfrac{a-b}{\sqrt{ab}-b}$ $=\lim_{b\rightarrow{a}}(\sqrt{ab} \dfrac{a}{a+\sqrt{ab}})×\dfrac{\color{#E81990}{(\sqrt{a}-\sqrt{b})}\color{#333333}(\sqrt{a}+\sqrt{b})}{\sqrt{a}\color{#E81990}{(\sqrt{a}-\sqrt{b})}}$ $\implies\boxed{a}$