Prove that `lim_(xto2) [x]` does not exists, where [.] represents the greatest integer function.

Explanation

To prove that \(\lim_{x \to 2} [x]\) does not exist, where \([x]\) represents the greatest integer function, we will analyze the left-hand limit and the right-hand limit as \(x\) approaches 2.

### Step-by-Step Solution:

1. **Define the Greatest Integer Function**: The greatest integer function \([x]\) gives the largest integer less than or equal to \(x\). For example, \([2.5] = 2\) and \([2] = 2\).

2. **Calculate the Left-Hand Limit**:
- We need to find \(\lim_{x \to 2^-} [x]\).
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