Let `f(x)=[x]=` Greatest integer less than or equal to x and k be an integer. Then, which one of the following in not correct?

Options

A. `lim_(xtok^-)f(x)=k-1`
B. `lim _(xtok)f(x)=k`
C. `lim _(xtok)f(x)"exists"`
D. `lim_(xtok)f(x)` does not exist

Explanation

To solve the problem, we need to analyze the function \( f(x) = [x] \), which represents the greatest integer less than or equal to \( x \). We will evaluate the limits as \( x \) approaches an integer \( k \) from both sides (left and right) and determine which of the provided options is incorrect.

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) = [x] \) gives the greatest integer less than or equal to \( x \). For example:
- \( f(2.5) = 2 \)
- \( f(3) = 3 \)
- \( f(3.999) = 3 \)

2. **Evaluating the First Option**:
\[ \lim_{x \to k^-} f(x) = \lim_{x \to k^-} [x] \]
As \( x \) approaches \( k \) from the left (i.e., \( k - \epsilon \) where \( \epsilon \) is a small positive number), \( f(x) \) will be \( k - 1 \) because \( k - \epsilon \) is less than \( k \) but greater than \( k - 1 \).
- Therefore, \( \lim_{x \to k^-} f(x) = k - 1 \).
- This option is **correct**.

3. **Evaluating the Second Option**:
\[ \lim_{x \to k} f(x) = \lim_{x \to k} [x] \]
When \( x \) is exactly \( k \), \( f(k) = k \).
- Therefore, \( \lim_{x \to k} f(x) = k \).
- This option is **correct**.

4. **Evaluating the Third Option**:
We need to check if the limit exists:
- **Left-hand limit**:
\[ \lim_{x \to k^-} f(x) = k - 1 \]
- **Right-hand limit**:
\[ \lim_{x \to k^+} f(x) = \lim_{x \to k^+} [x] = k \]
Since the left-hand limit \( (k - 1) \) is not equal to the right-hand limit \( (k) \), the overall limit does not exist.
- Therefore, this option is **incorrect**.

5. **Evaluating the Fourth Option**:
The statement says that the limit does not exist, which we have just established is true.
- Therefore, this option is **correct**.

### Conclusion:
The only incorrect statement is the third option, which claims that the limit exists.