`Lt_(xto2)` [x] where `[*]` denotes the greatest integer function is equal to

Options

A. A. 2
B. B. 1
C. C. 0
D. D. does not exist

Answer

To solve the limit \( \lim_{x \to 2} [x] \) where \([x]\) denotes the greatest integer function, we will analyze the left-hand limit (LHL) and the right-hand limit (RHL) as \( x \) approaches 2.

### Step 1: Find the Left-Hand Limit (LHL)
The left-hand limit as \( x \) approaches 2 is denoted as:
\[ \lim_{x \to 2^-} [x] \]
As \( x \) approaches 2 from the left (values slightly less than 2), we can consider values like \( 1.9, 1.99, 1.999 \), etc. The greatest integer function \([x]\) gives us:
- For \( x = 1.9 \), \([1.9] = 1\)
- For \( x = 1.99 \), \([1.99] = 1\)
- For \( x = 1.999 \), \([1.999] = 1\)

Thus, we can conclude:
\[ \lim_{x \to 2^-} [x] = 1 \]

### Step 2: Find the Right-Hand Limit (RHL)
The right-hand limit as \( x \) approaches 2 is denoted as:
\[ \lim_{x \to 2^+} [x] \]
As \( x \) approaches 2 from the right (values slightly greater than 2), we can consider values like \( 2.1, 2.01, 2.001 \), etc. The greatest integer function \([x]\) gives us:
- For \( x = 2.1 \), \([2.1] = 2\)
- For \( x = 2.01 \), \([2.01] = 2\)
- For \( x = 2.001 \), \([2.001] = 2\)

Thus, we can conclude:
\[ \lim_{x \to 2^+} [x] = 2 \]

### Step 3: Compare the Limits
Now we compare the left-hand limit and the right-hand limit:
\[ \lim_{x \to 2^-} [x] = 1 \quad \text{and} \quad \lim_{x \to 2^+} [x] = 2 \]
Since the left-hand limit is not equal to the right-hand limit, we conclude that:
\[ \lim_{x \to 2} [x] \text{ does not exist.} \]

### Final Answer
The limit \( \lim_{x \to 2} [x] \) does not exist.

### Options
- a) 2
- b) 1
- c) 0
- d) does not exist

The correct option is **d) does not exist**.