If `f(x)=|x-1|-[x]`, where `[x]` is the greatest integer less than or equal to x, then

Options

A. `underset(xto0)lim[f(x)]=0`
B. `underset(xto0)lim[f(x)]=1`
C. `underset(xto0)lim[(f(x))/(x)]` does not exist
D. `underset(xto0)lim[(f(x))/(x)]` exists

Explanation

To solve the limit problem given the function \( f(x) = |x - 1| - [x] \) where \([x]\) is the greatest integer less than or equal to \(x\), we will find the right-hand limit and the left-hand limit as \(x\) approaches 1.

### Step 1: Find the Right-Hand Limit
We need to evaluate the limit as \(x\) approaches 1 from the right, denoted as \( \lim_{x \to 1^+} f(x) \).

1. **Substitute \(x = 1 + h\)** where \(h\) approaches 0 from the positive side (i.e., \(h \to 0^+\)).
2. Then, \(f(1 + h) = |(1 + h) - 1| - [1 + h]\).
3. This simplifies to \(f(1 + h) = |h| - [1 + h]\).
4. Since \(h\) is positive, \(|h| = h\).
5. The greatest integer function \([1 + h]\) will equal 1 because \(1 + h\) is between 1 and 2.
6. Therefore, \(f(1 + h) = h - 1\).
7. Now, taking the limit as \(h \to 0^+\):
\[ \lim_{h \to 0^+} (h - 1) = 0 - 1 = -1. \]

### Step 2: Find the Left-Hand Limit
Next, we evaluate the limit as \(x\) approaches 1 from the left, denoted as \( \lim_{x \to 1^-} f(x) \).

1. **Substitute \(x = 1 - h\)** where \(h\) approaches 0 from the positive side (i.e., \(h \to 0^+\)).
2. Then, \(f(1 - h) = |(1 - h) - 1| - [1 - h]\).
3. This simplifies to \(f(1 - h) = | - h| - [1 - h]\).
4. Since \(h\) is positive, \(| - h| = h\).
5. The greatest integer function \([1 - h]\) will equal 0 because \(1 - h\) is between 0 and 1.
6. Therefore, \(f(1 - h) = h - 0 = h\).
7. Now, taking the limit as \(h \to 0^+\):
\[ \lim_{h \to 0^+} h = 0. \]

### Step 3: Compare the Limits
Now we compare the right-hand limit and the left-hand limit:
- Right-hand limit: \( \lim_{x \to 1^+} f(x) = -1 \)
- Left-hand limit: \( \lim_{x \to 1^-} f(x) = 0 \)

Since the right-hand limit and left-hand limit are not equal, we conclude that:
\[ \lim_{x \to 1} f(x) \text{ does not exist.} \]

### Final Answer
The limit \( \lim_{x \to 1} f(x) \) does not exist.