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The limit provided is:

\[
\lim_{{x \to 0}} \frac{x}{|x|}
\]

### Solution:

We need to analyze the behavior of this expression as \( x \to 0 \), which requires us to break it into two cases based on the definition of the absolute value function \( |x| \).

1. **When \( x > 0 \):**
   - In this case, \( |x| = x \). 
   \[
   \frac{x}{|x|} = \frac{x}{x} = 1
   \]
   Therefore, as \( x \to 0^+ \) (from the right), the limit is:
   \[
   \lim_{{x \to 0^+}} \frac{x}{|x|} = 1
   \]

2. **When \( x < 0 \):**
   - In this case, \( |x| = -x \).
   \[
   \frac{x}{|x|} = \frac{x}{-x} = -1
   \]
   Therefore, as \( x \to 0^- \) (from the left), the limit is:
   \[
   \lim_{{x \to 0^-}} \frac{x}{|x|} = -1
   \]

### Conclusion:

The left-hand limit as \( x \to 0^- \) is \( -1 \), and the right-hand limit as \( x \to 0^+ \) is \( 1 \). Since these two limits do not match, the overall limit does not exist.

So:
\[
\lim_{{x \to 0}} \frac{x}{|x|} \text{ does not exist}.
\]

Would you like further details or have any questions?

### Related Questions:
1. What is the formal definition of a limit not existing at a point?
2. How does the absolute value function impact limits in general?
3. What happens if the function inside the limit was \( \frac{|x|}{x} \)?
4. Can the limit of piecewise functions exist if one-sided limits differ?
5. How do you compute limits approaching infinity involving absolute values?

### Tip:
When working with absolute value functions, always consider breaking the analysis into cases where \( x \) is positive and where \( x \) is negative.

Evaluate the limit: lim (x → 0) [x / |x|]

Learn how to evaluate the limit lim (x → 0) [x / |x|]. This problem involves key mathematical concepts such as limits, absolute value, and piecewise functions. Detailed step-by-step solution provided for students in Grades 11-12.

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