Evaluate : `lim_(x to -1) x/([x])`

Answer

To evaluate the limit \( \lim_{x \to -1} \frac{x}{[x]} \), where \([x]\) denotes the greatest integer function (also known as the floor function), we will analyze the behavior of the function as \(x\) approaches \(-1\) from both the left and the right.

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**:
The greatest integer function \([x]\) returns the largest integer less than or equal to \(x\). For example:
- If \(x = -1.5\), then \([-1.5] = -2\).
- If \(x = -1.0\), then \([-1.0] = -1\).
- If \(x = -0.5\), then \([-0.5] = -1\).

2. **Finding the Right-Hand Limit**:
We first compute the right-hand limit as \(x\) approaches \(-1\) from the right (\(x \to -1^+\)):
\[ \lim_{x \to -1^+} \frac{x}{[x]} \]
As \(x\) approaches \(-1\) from the right, \(x\) is slightly greater than \(-1\) (e.g., \(-0.999\)). Therefore, \([x] = -1\).
\[ \lim_{x \to -1^+} \frac{x}{[x]} = \lim_{x \to -1^+} \frac{x}{-1} = \lim_{x \to -1^+} -x = -(-1) = 1 \]

3. **Finding the Left-Hand Limit**:
Next, we compute the left-hand limit as \(x\) approaches \(-1\) from the left (\(x \to -1^-\)):
\[ \lim_{x \to -1^-} \frac{x}{[x]} \]
As \(x\) approaches \(-1\) from the left, \(x\) is slightly less than \(-1\) (e.g., \(-1.001\)). Therefore, \([x] = -2\).
\[ \lim_{x \to -1^-} \frac{x}{[x]} = \lim_{x \to -1^-} \frac{x}{-2} = \lim_{x \to -1^-} -\frac{x}{2} = -\frac{-1}{2} = \frac{1}{2} \]

4. **Comparing the Limits**:
We have:
- Right-hand limit: \(1\)
- Left-hand limit: \(\frac{1}{2}\)

Since the left-hand limit and the right-hand limit are not equal, we conclude that the limit does not exist.

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