The given limit is:

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

To solve this, let's consider the behavior of \(\frac{|x|}{x}\) as \(x\) approaches 0 from both the positive and negative sides.

1. **When \(x > 0\):**
   \[
   \frac{|x|}{x} = \frac{x}{x} = 1
   \]
   So, as \(x \to 0^+\) (from the right-hand side), \(\frac{|x|}{x} = 1\).

2. **When \(x < 0\):**
   \[
   \frac{|x|}{x} = \frac{-x}{x} = -1
   \]
   So, as \(x \to 0^-\) (from the left-hand side), \(\frac{|x|}{x} = -1\).

### Conclusion:
Since the left-hand limit (\(-1\)) does not equal the right-hand limit (\(1\)), the limit **does not exist**.

Would you like more details on this explanation or have any questions?

Here are 5 related questions to explore:

1. What is the formal definition of a limit?
2. How does the concept of one-sided limits help in solving piecewise functions?
3. What are some other functions where limits behave differently from each side?
4. How can we find the limit of a function involving absolute values in more complex expressions?
5. How does the epsilon-delta definition of limits apply to functions like \(\frac{|x|}{x}\)?

**Tip:** When dealing with absolute value functions, always consider breaking down the limit into one-sided limits for better clarity!

Explore the limit of |x|/x as x approaches 0 and learn why the limit does not exist. This problem covers key concepts such as one-sided limits and absolute value functions, making it suitable for high school or introductory college calculus students.

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