https://r2.erweima.ai/mathbot/image/1726187331073ivxnie2k.jpg

The function presented in the image is:

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

You are asked to determine which of the options corresponds to the values of the function as \(x \to 3^-\) (approaching 3 from the left) and \(x \to 3^+\) (approaching 3 from the right).

### Step 1: Analyze the behavior of the function as \(x \to 3\)
At \(x = 3\), the function has the form:
\[
f(3) = \frac{9 - 3^2}{3 - 3} = \frac{9 - 9}{0} = \frac{0}{0}
\]
This is an indeterminate form, so we need to simplify the expression to understand its behavior near \(x = 3\).

### Step 2: Simplify the function
We factor the numerator:
\[
9 - x^2 = (3 - x)(3 + x)
\]
Substituting this into the original expression:
\[
f(x) = \frac{(3 - x)(3 + x)}{3 - x}
\]
For \(x \neq 3\), we can cancel the term \(3 - x\) from the numerator and denominator:
\[
f(x) = 3 + x
\]

### Step 3: Evaluate limits as \(x \to 3\)
Now, we evaluate the limit of \(f(x)\) as \(x\) approaches 3 from both sides:
- As \(x \to 3\), \(f(x) = 3 + 3 = 6\).

Thus, the function approaches 6 as \(x\) gets closer to 3 from both the left and right. The correct option corresponds to values near 6.

### Step 4: Compare with the options
The option that reflects values near 6 is:
- Option **a**, which starts with 6.01 and gradually approaches 6.000001 as \(x\) approaches 3.

### Final Answer:
The correct choice is **a**.

Would you like more details or have any questions?

### Related Questions:
1. What happens if we couldn't cancel out the factor \(3 - x\)?
2. How would the function behave if the numerator was not factorable?
3. How do limits behave with rational functions in general?
4. Can indeterminate forms always be resolved by simplification?
5. How does factoring help in limit calculations?

### Tip:
When you encounter an indeterminate form, always try simplifying by factoring or applying L'Hopital's Rule for further analysis.

Considere la función f(x) = (9 - x^2) / (3 - x). ¿Cuál de las siguientes opciones corresponde a los valores de la función cuando x → 3?

Learn how to determine the limit of the rational function f(x) = (9 - x^2) / (3 - x) as x approaches 3. This solution involves simplifying an indeterminate form using factoring techniques and evaluating the limit.

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