$y=\lim _{x \rightarrow-3} \frac{\sqrt{x+7}-2}{x^{2}-9}$
\(\boxed{-\frac{1}{24}}\)
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 :\(\boxed{-\frac{1}{24}}\)