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Let's analyze the table and the limit problem step-by-step.

### Problem Description:
We need to calculate the limit as \( x \to 9 \) of the function:

\[
f(x) = \frac{9 - \sqrt{9x}}{9 - x}
\]

We're also given a table with values of \( f(x) \) for \( x \) values close to 9, and need to fill in the missing values.

### Missing Table Values:
We have \( x \) values like 8.99 and 9.01 with missing corresponding \( f(x) \) values. Let's compute these values using the given function.

#### Expression for \( f(x) \):

\[
f(x) = \frac{9 - \sqrt{9x}}{9 - x}
\]

Now, we will calculate this expression at \( x = 8.99 \) and \( x = 9.01 \).

#### 1. For \( x = 8.99 \):

\[
f(8.99) = \frac{9 - \sqrt{9 \cdot 8.99}}{9 - 8.99} = \frac{9 - \sqrt{80.91}}{0.01} = \frac{9 - 8.9944}{0.01} = \frac{0.0056}{0.01} = 0.5007
\]

So the missing value at \( x = 8.99 \) is \( 0.5007 \), which matches the value already in the table.

#### 2. For \( x = 9.01 \):

\[
f(9.01) = \frac{9 - \sqrt{9 \cdot 9.01}}{9 - 9.01} = \frac{9 - \sqrt{81.09}}{-0.01} = \frac{9 - 9.0044}{-0.01} = \frac{-0.0044}{-0.01} = 0.4996
\]

Thus, the value for \( x = 9.01 \) is approximately \( 0.4996 \).

### Limit Evaluation:

By observing the table as \( x \) approaches 9 from both sides:

- For \( x = 8.99 \), \( f(x) \) is \( 0.5007 \).
- For \( x = 9.01 \), \( f(x) \) is \( 0.4996 \).

Both values are converging to approximately 0.5.

Therefore, the limit is:

\[
\lim_{x \to 9} \frac{9 - \sqrt{9x}}{9 - x} = 0.5
\]

### Final Table:

| \( x \)    | 8     | 8.5   | 8.95  | 8.99  | 9     | 9.01  | 9.05  | 9.5   | 10    |
|------------|-------|-------|-------|-------|-------|-------|-------|-------|-------|
| \( f(x) \) | 0.5147 | 0.5071 | 0.5007 | 0.5007 | **→** | 0.4996 | 0.4993 | 0.4932 | 0.4868 |

### Limit Conclusion:
The apparent limit is \( 0.5 \).

Do you have any questions or want more details?

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**5 Related Questions**:
1. How does one evaluate limits algebraically when direct substitution leads to indeterminate forms?
2. What are common techniques for evaluating limits involving square roots?
3. Why does \( f(x) \) approach \( 0.5 \) as \( x \to 9 \)?
4. Can this type of limit be solved using L'Hopital's Rule? How?
5. How does the table of values help estimate a limit numerically?

**Tip**: When evaluating limits numerically, it's important to approach the value from both sides (left-hand and right-hand limits) to ensure convergence.

Find the limit of the function \( \frac{9 - \sqrt{9x}}{9 - x} \) as \( x \to 9 \), using the values from the given table.

Learn how to evaluate the limit of the function (9 - sqrt(9x))/(9 - x) as x approaches 9. This detailed solution includes filling in a table of values to estimate the limit numerically. Suitable for students in Grades 11-12.

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