Evaluate the following limits :
`Lim_(xto3^(-)) ([x]-x)`

Answer

To evaluate the limit \( \lim_{x \to 3^-} ([x] - x) \), where \([x]\) is the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Understand the Greatest Integer Function
The greatest integer function \([x]\) returns the largest integer less than or equal to \(x\). For example:
- If \(x = 2.5\), then \([x] = 2\).
- If \(x = 3\), then \([x] = 3\).

### Step 2: Analyze the Limit as \(x\) Approaches 3 from the Left
Since we are evaluating the limit as \(x\) approaches 3 from the left (denoted \(3^-\)), we consider values of \(x\) that are slightly less than 3, such as \(2.9\), \(2.99\), etc.

For any \(x\) in the interval \(2 < x < 3\):
- The greatest integer function \([x] = 2\).

### Step 3: Substitute into the Limit Expression
Now we can substitute this into the limit expression:
\[ \lim_{x \to 3^-} ([x] - x) = \lim_{x \to 3^-} (2 - x) \]

### Step 4: Evaluate the Limit
As \(x\) approaches 3 from the left, we can directly substitute \(x = 3\) into the expression \(2 - x\):
\[ \lim_{x \to 3^-} (2 - x) = 2 - 3 = -1 \]

### Final Answer
Thus, the value of the limit is:
\[ \boxed{-1} \]