To evaluate the limit as $x$ approaches 3 for the expression $\frac{\sqrt{x^2 - 7}}{\sqrt{x} - 3}$, we can proceed as follows:

Step 1: Direct Substitution

First, try substituting $x = 3$ directly into the expression:

$\frac{\sqrt{3^2 - 7}}{\sqrt{3} - 3} = \frac{\sqrt{9 - 7}}{\sqrt{3} - 3} = \frac{\sqrt{2}}{\sqrt{3} - 3}$

This is not an indeterminate form, so the limit can be evaluated directly.

Step 2: Evaluate the Expression

Now, we evaluate the expression with the substituted value:

$\frac{\sqrt{2}}{\sqrt{3} - 3}$

This is the simplified form, so the limit as $x$ approaches 3 is:

$\lim_{{x \to 3}} \frac{\sqrt{x^2 - 7}}{\sqrt{x} - 3} = \frac{\sqrt{2}}{\sqrt{3} - 3}$

Final Answer:

The limit is $\boxed{\frac{\sqrt{2}}{\sqrt{3} - 3}}$.

Would you like more details on this solution, or have any questions?

Here are five related questions:

1. How do you evaluate limits that result in indeterminate forms?
2. What is the significance of rationalizing the denominator?
3. Can you apply L'Hôpital's Rule to solve this limit problem?
4. How do square roots affect the calculation of limits?
5. What are some common techniques for solving limits involving radicals?

Tip: Always try direct substitution first when solving limits, as it can immediately give you the result if the expression is not indeterminate.