`Lt_(x to (3)/(2))[x]` is equal to (i) 1 (ii) `-1` (iii) 2 (iv) does not exist

Options

A. 1
B. `-1`
C. 2
D. does not exist

Answer

To solve the limit \( \lim_{x \to \frac{3}{2}} [x] \), where \([x]\) denotes the greatest integer function (also known as the floor function), we will analyze the left-hand limit and the right-hand limit.

### Step-by-Step Solution:

1. **Identify the point of interest**: We are looking at the limit as \( x \) approaches \( \frac{3}{2} \) (which is 1.5).

2. **Calculate the left-hand limit**:
\[ \lim_{x \to \frac{3}{2}^-} [x] \]
As \( x \) approaches \( \frac{3}{2} \) from the left (values slightly less than 1.5), the greatest integer function \([x]\) will take the value of the greatest integer less than 1.5, which is 1.
\[ \lim_{x \to \frac{3}{2}^-} [x] = 1 \]

3. **Calculate the right-hand limit**:
\[ \lim_{x \to \frac{3}{2}^+} [x] \]
As \( x \) approaches \( \frac{3}{2} \) from the right (values slightly more than 1.5), the greatest integer function \([x]\) will take the value of the greatest integer less than or equal to 1.5, which is 2.
\[ \lim_{x \to \frac{3}{2}^+} [x] = 2 \]

4. **Compare the left-hand limit and right-hand limit**:
- Left-hand limit: \( 1 \)
- Right-hand limit: \( 2 \)

5. **Conclusion**: Since the left-hand limit is not equal to the right-hand limit, the overall limit does not exist.
\[ \lim_{x \to \frac{3}{2}} [x] \text{ does not exist.} \]

### Final Answer:
The correct option is (iv) does not exist.