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In this problem, you are given the function:

\[
f(x) = \frac{x - 3}{x^2 - 9}
\]

and asked to complete the table of values as \(x\) approaches 3. You are also asked to estimate the limit as \(x\) approaches 3 using the function's behavior near that point.

Let's walk through the steps:

1. **Simplify the function**:
   The denominator \(x^2 - 9\) can be factored:
   \[
   x^2 - 9 = (x - 3)(x + 3)
   \]
   So the function becomes:
   \[
   f(x) = \frac{x - 3}{(x - 3)(x + 3)} = \frac{1}{x + 3}, \quad \text{for} \quad x \neq 3
   \]
   This is valid for all \(x \neq 3\), as dividing by zero would otherwise occur at \(x = 3\).

2. **Estimating the limit**:
   As \(x \to 3\), the function simplifies to:
   \[
   \lim_{x \to 3} f(x) = \lim_{x \to 3} \frac{1}{x + 3} = \frac{1}{6} \approx 0.1667
   \]
   This is the value you are asked to enter in the table where \(x = 3\), which is also verified with the limit at the bottom.

3. **Table values**:
   - You are filling in values of \(f(x)\) for \(x = 2.9, 2.999, 3.001, 3.01, 3.1\), and the values are already calculated for some entries:
     - \(x = 2.9\), \(f(x) = 0.1695\)
     - \(x = 2.999\), \(f(x) = 0.1667\)
     - \(x = 3.001\), \(f(x) = 0.1667\)
     - \(x = 3.01\), \(f(x) = 0.1669\)
     - \(x = 3.1\), \(f(x) = 0.4762\)

4. **Conclusion**:
   The value of the limit as \(x \to 3\) is \( \lim_{x \to 3} f(x) = 0.1667 \).

Would you like to see a detailed explanation of any step, or further clarification?

### Follow-up Questions:
1. How does factoring help simplify the given function?
2. What happens to the function when \(x = 3\)?
3. Why do we check values around \(x = 3\) to estimate the limit?
4. Can you explain why \(f(x)\) is undefined at \(x = 3\)?
5. How does the limit behave as \(x\) approaches values slightly larger than 3?

### Tip:
Always simplify functions as much as possible before calculating limits to avoid unnecessary complications.

Complete the table for the function f(x) = (x - 3) / (x^2 - 9) and estimate the limit as x approaches 3.

Learn how to simplify and calculate the limit of the rational function f(x) = (x - 3) / (x^2 - 9) as x approaches 3. This step-by-step guide explores factoring, function behavior, and limits, perfect for students in Grades 10-12.

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