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The problem presents a function \( f(x) = \frac{9 - x^2}{3 - x} \) and asks which of the given options corresponds to the values of the function as \( x \to 3^- \) (from the left side) and \( x \to 3^+ \) (from the right side).

### Step 1: Analyze the function

The function is:
\[
f(x) = \frac{9 - x^2}{3 - x}
\]
We can simplify the numerator:
\[
9 - x^2 = (3 - x)(3 + x)
\]
Thus, the function becomes:
\[
f(x) = \frac{(3 - x)(3 + x)}{3 - x}
\]
As long as \( x \neq 3 \), we can cancel out the \( (3 - x) \) terms:
\[
f(x) = 3 + x \quad \text{for} \quad x \neq 3
\]

### Step 2: Behavior as \( x \to 3 \)

As \( x \) approaches 3 from both sides, the function simplifies to:
\[
f(x) = 3 + 3 = 6
\]
So, the limit as \( x \to 3 \) is 6. 

### Step 3: Check the given options

From the table, the values approaching 6 are in option (a), which shows values converging to 6.01, 6.001, 6.0001, and so on, as \( x \to 3 \).

### Conclusion:
The correct answer is **option (a)**.

Would you like more details or have any questions about this?

Here are some related questions:
1. How does the concept of limits help in understanding the behavior of functions?
2. What happens if a function cannot be simplified directly?
3. Can a function have different limits from the left and right at the same point?
4. What is the formal definition of a limit as \( x \to a \)?
5. How can we detect removable discontinuities in a function?

**Tip**: Always try to simplify a function before taking limits to identify potential removable discontinuities or singularities.

Consider the function f(x) = (9 - x^2) / (3 - x). Which of the following options corresponds to the values of the function as x approaches 3 from the left and right?

Discover how to analyze the limit of the function f(x) = (9 - x^2) / (3 - x) as x approaches 3 from both directions, with simplifications and detailed explanations on discontinuities and behavior near critical points. Suitable for high school students studying calculus.

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