$y=\lim _{x \rightarrow-3} \frac{\sqrt{x+7}-2}{x^{2}-9}$

Step 1 ：Given the function: \(y=\frac{\sqrt{x+7}-2}{x^{2}-9}\)

Step 2 ：First, we need to simplify the expression. We can use the conjugate method by multiplying the numerator and denominator by \(\sqrt{x+7}+2\):

Step 3 ：\(y=\frac{\sqrt{x+7}-2}{x^{2}-9} \cdot \frac{\sqrt{x+7}+2}{\sqrt{x+7}+2}\)

Step 4 ：\(y=\frac{(x+7)-4}{(x+3)(x-3)(\sqrt{x+7}+2)}\)

Step 5 ：\(y=\frac{x+3}{(x+3)(x-3)(\sqrt{x+7}+2)}\)

Step 6 ：Now, we can cancel out the \((x+3)\) terms:

Step 7 ：\(y=\frac{1}{(x-3)(\sqrt{x+7}+2)}\)

Step 8 ：Finally, substitute x = -3 to find the limit:

Step 9 ：\(y=\frac{1}{(-6)(\sqrt{-3+7}+2)}\)

Step 10 ：\(y=\frac{1}{(-6)(2+2)}\)

Step 11 ：\(y=-\frac{1}{24}\)

Step 12 ：\(\boxed{-\frac{1}{24}}\)