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The limit problem you provided is:

\[
\lim_{{x \to 1}} \frac{|x - 1|}{x - 1}
\]

### Analysis of the Limit

1. **Absolute value function:**
   The expression involves the absolute value of \( x - 1 \). The absolute value function behaves differently depending on whether \( x \) is greater than or less than 1.
   
2. **Piecewise definition of \( |x - 1| \):**
   - For \( x > 1 \), \( |x - 1| = x - 1 \).
   - For \( x < 1 \), \( |x - 1| = -(x - 1) \).

### Evaluating the limit:

We need to consider the left-hand limit (\( x \to 1^- \)) and the right-hand limit (\( x \to 1^+ \)) separately.

#### 1. Right-hand limit (\( x \to 1^+ \)):
- When \( x > 1 \), \( |x - 1| = x - 1 \), so the expression becomes:
  \[
  \lim_{{x \to 1^+}} \frac{x - 1}{x - 1} = \lim_{{x \to 1^+}} 1 = 1
  \]

#### 2. Left-hand limit (\( x \to 1^- \)):
- When \( x < 1 \), \( |x - 1| = -(x - 1) \), so the expression becomes:
  \[
  \lim_{{x \to 1^-}} \frac{-(x - 1)}{x - 1} = \lim_{{x \to 1^-}} -1 = -1
  \]

### Conclusion:

- The right-hand limit as \( x \to 1^+ \) is 1.
- The left-hand limit as \( x \to 1^- \) is -1.

Since the left-hand and right-hand limits are not equal, the overall limit **does not exist**.

If you have any questions or want further details, feel free to ask!

### Related Questions:
1. What are the conditions for a limit to exist?
2. How does the absolute value function affect limits?
3. What is the difference between left-hand and right-hand limits?
4. Can you find a limit involving other piecewise functions?
5. How do limits behave when approaching infinity?

### Tip:
Always check for different behaviors on both sides of the point where a function involves an absolute value.

Evaluate the limit: lim(x→1) |x−1|/(x−1)

Learn how to evaluate the limit lim(x→1) |x−1|/(x−1) with step-by-step analysis. This problem focuses on the behavior of absolute value and piecewise functions as limits approach a point from both sides. Suitable for high school students in Grades 10-12.

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