We need to evaluate the limit of the given expression as $x$ approaches 3: $$ \lim_{x\to 3} \frac{9-x^2}{-34 - \sqrt{x^2+7}} $$

1. Problem Description

We need to evaluate the limit of the given expression as $x$ approaches 3:

$$$$

\lim_{x\to 3} \frac{9-x^2}{-34 - \sqrt{x^2+7}}

$$$$

2. Solution Steps

First, let's directly substitute $x=3$ into the expression:

$$$$

\frac{9 - (3)^2}{-34 - \sqrt{(3)^2 + 7}} = \frac{9-9}{-34 - \sqrt{9+7}} = \frac{0}{-34 - \sqrt{16}} = \frac{0}{-34 - 4} = \frac{0}{-38} = 0

$$$$

Since the direct substitution yields 0, we do not have an indeterminate form, and the limit is simply

0.

3. Final Answer

0