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.