\[\frac{b - 9}{b^{\frac{1}{2}} + 3} = \frac{\left( b^{\frac{1}{2}} + 3 \right)\left( b^{\frac{1}{2}} - 3 \right)}{b^{\frac{1}{2}} + 3} =\]
\[= b^{\frac{1}{2}} - 3 = \sqrt{b} - 3.\]