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Let's analyze and solve the given limit problem step by step, following parts (a), (b), and (c).

We are asked to find the limit:

\[
\lim_{{x \to 1}} f(x), \text{ where } f(x) = \frac{1 - \sqrt{x}}{x - 1}
\]

### Part (a): Complete the Table to Estimate the Limit

We need to estimate the limit by evaluating \( f(x) \) for values of \( x \) close to 1 (like 1.1, 1.01, 1.001, and 1.0001).

We will calculate \( f(x) \) for these values of \( x \).

Let me compute those values first.For part (a), the values of \( f(x) \) corresponding to the given values of \( x \) are:

| \( x \)    | \( f(x) \)          |
|------------|---------------------|
| 1.1        | -0.4881             |
| 1.01       | -0.4988             |
| 1.001      | -0.4999             |
| 1.0001     | -0.49999            |

As \( x \) approaches 1, the values of \( f(x) \) seem to approach \(-0.5\).

### Part (b): Use a Graphing Utility to Estimate the Limit

By examining the trend from the table, it appears that \( \lim_{{x \to 1}} f(x) = -0.5 \).

### Part (c): Rationalize the Numerator

To analytically find the limit, we rationalize the numerator of the function:

\[
f(x) = \frac{1 - \sqrt{x}}{x - 1}
\]

Multiply the numerator and the denominator by \( 1 + \sqrt{x} \):

\[
f(x) = \frac{(1 - \sqrt{x})(1 + \sqrt{x})}{(x - 1)(1 + \sqrt{x})}
\]

Simplifying the numerator:

\[
(1 - \sqrt{x})(1 + \sqrt{x}) = 1^2 - (\sqrt{x})^2 = 1 - x
\]

Thus, the function becomes:

\[
f(x) = \frac{1 - x}{(x - 1)(1 + \sqrt{x})} = \frac{-(x - 1)}{(x - 1)(1 + \sqrt{x})}
\]

Canceling out \( x - 1 \) from the numerator and denominator:

\[
f(x) = \frac{-1}{1 + \sqrt{x}}
\]

Now, take the limit as \( x \to 1 \):

\[
\lim_{{x \to 1}} f(x) = \frac{-1}{1 + \sqrt{1}} = \frac{-1}{1 + 1} = \frac{-1}{2}
\]

Thus, the exact value of the limit is:

\[
\lim_{{x \to 1}} f(x) = -\frac{1}{2}
\]

This confirms the estimate from parts (a) and (b).

---

Do you need further clarification or details on any part? Here are five related questions to explore further:

1. What happens if we change the function to \( f(x) = \frac{\sqrt{x} - 1}{x - 1} \)?
2. How does rationalizing the numerator help in solving limits?
3. Can you explain how limits work when the function is undefined at a point?
4. What is the role of continuity in evaluating limits algebraically?
5. How can graphical estimations of limits be improved using technology?

**Tip:** When you see a square root in a limit problem, rationalizing the numerator is a common strategy to simplify and solve it analytically.

Learn how to solve a calculus problem involving limits by rationalizing the numerator. This detailed solution covers the limit of the function \( f(x) = \frac{1 - \sqrt{x}}{x - 1} \) as x approaches 1. Suitable for high school students in Grades 11-12.

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