To find the limit:

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

Here, \([x]\) denotes the greatest integer function, or the floor function, which gives the largest integer less than or equal to \(x\).

### Step-by-step Solution:

1. **Case 1: Approaching from the right (\(x \to 1^+\)):**
   - If \(x\) is slightly greater than 1, then \([x] = 1\).
   - The expression becomes:
   \[
   \frac{1}{x}
   \]
   - As \(x \to 1^+\), \(\frac{1}{x} \to \frac{1}{1} = 1\).

2. **Case 2: Approaching from the left (\(x \to 1^-\)):**
   - If \(x\) is slightly less than 1, then \([x] = 0\).
   - The expression becomes:
   \[
   \frac{0}{x} = 0
   \]
   - As \(x \to 1^-\), the expression evaluates to 0.

### Conclusion:
The left-hand limit as \(x \to 1^-\) is 0, and the right-hand limit as \(x \to 1^+\) is 1. Since the left-hand and right-hand limits are not equal, the limit does not exist.

### Final Answer:
\[
\lim_{{x \to 1}} \frac{{[x]}}{x} \text{ does not exist.}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the definition of the greatest integer function?
2. How do you compute limits with piecewise functions?
3. What are left-hand and right-hand limits?
4. How do you prove that a limit does not exist?
5. What is the relationship between continuity and limits?

**Tip:** Always check both left-hand and right-hand limits when solving limits involving piecewise or floor functions.

This webpage provides a detailed solution to the limit problem lim (x → 1) [x]/x, involving the greatest integer (floor) function. Explore step-by-step solutions for left-hand and right-hand limits, and understand why the overall limit does not exist. Suitable for advanced high school students (Grades 11-12).

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