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The given problem is to find the limit:

\[
\lim_{x \to 3} \frac{\sqrt{x} - \sqrt{3} + \sqrt{x - 3}}{\sqrt{x^2 - 9}}
\]

### Step 1: Simplify the denominator
The denominator \(\sqrt{x^2 - 9}\) can be rewritten using the difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
So,
\[
\sqrt{x^2 - 9} = \sqrt{(x - 3)(x + 3)} = \sqrt{x - 3} \cdot \sqrt{x + 3}
\]

### Step 2: Substituting and simplifying the expression
Now, substitute this into the limit:

\[
\lim_{x \to 3} \frac{\sqrt{x} - \sqrt{3} + \sqrt{x - 3}}{\sqrt{x - 3} \cdot \sqrt{x + 3}}
\]

### Step 3: Check for indeterminate form
At \(x = 3\), both the numerator and denominator approach zero, indicating an indeterminate form \(\frac{0}{0}\). This suggests that further simplification is needed, possibly by rationalizing the numerator.

### Step 4: Rationalize the numerator
We can try to simplify the numerator by grouping and rationalizing terms. To simplify \(\sqrt{x} - \sqrt{3}\), multiply by the conjugate:

\[
\sqrt{x} - \sqrt{3} = \frac{(\sqrt{x} - \sqrt{3})(\sqrt{x} + \sqrt{3})}{\sqrt{x} + \sqrt{3}} = \frac{x - 3}{\sqrt{x} + \sqrt{3}}
\]

Thus, the expression becomes:

\[
\lim_{x \to 3} \frac{\frac{x - 3}{\sqrt{x} + \sqrt{3}} + \sqrt{x - 3}}{\sqrt{x - 3} \cdot \sqrt{x + 3}}
\]

### Step 5: Simplify further
Now, factor out \(\sqrt{x - 3}\) from the numerator:

\[
\lim_{x \to 3} \frac{\sqrt{x - 3} \left(\frac{1}{\sqrt{x} + \sqrt{3}} + 1\right)}{\sqrt{x - 3} \cdot \sqrt{x + 3}}
\]

Cancel out the \(\sqrt{x - 3}\) terms:

\[
\lim_{x \to 3} \frac{1}{\sqrt{x} + \sqrt{3}} + 1
\]

### Step 6: Substitute \(x = 3\)
Substituting \(x = 3\) into the remaining expression:

\[
\frac{1}{\sqrt{3} + \sqrt{3}} + 1 = \frac{1}{2\sqrt{3}} + 1 = \frac{1}{2\sqrt{3}} + \frac{2\sqrt{3}}{2\sqrt{3}} = \frac{1 + 2\sqrt{3}}{2\sqrt{3}}
\]

Thus, the final value of the limit is:

\[
\frac{1 + 2\sqrt{3}}{2\sqrt{3}}
\]

Would you like further details or have any questions about these steps?

### Related Questions:
1. How do you handle limits involving indeterminate forms?
2. What are other methods to simplify expressions involving square roots?
3. Can you explain the concept of conjugates in more depth?
4. How does the difference of squares formula help in limits?
5. What are other ways to solve limits involving radicals?

### Tip:
Always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) when evaluating limits and apply algebraic techniques like factoring or rationalization to resolve them.

Find the limit as x approaches 3 for the following expression: \( \frac{\sqrt{x} - \sqrt{3} + \sqrt{x - 3}}{\sqrt{x^2 - 9}} \)

Explore the step-by-step solution to find the limit of \( \frac{\sqrt{x} - \sqrt{3} + \sqrt{x - 3}}{\sqrt{x^2 - 9}} \) as x approaches 3. The solution involves algebraic techniques like factoring, rationalizing the numerator, and simplifying square roots.

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