Evaluate the following limits :
`Lim_( xto 5^(+)) (x - [x])`

Answer

To evaluate the limit \( \lim_{x \to 5^+} (x - [x]) \), 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 \([x]\) rounds down the real number \(x\) to the nearest integer less than or equal to \(x\). For example:
- \([1.2] = 1\)
- \([5.8] = 5\)
- \([5] = 5\)

### Step 2: Substitute \(x\) as \(5 + h\)
Since we are evaluating the limit as \(x\) approaches \(5\) from the right (denoted as \(5^+\)), we can express \(x\) as \(5 + h\), where \(h\) is a small positive number approaching \(0\).

### Step 3: Evaluate the Greatest Integer Function
For \(x = 5 + h\) (where \(h\) is a small positive number), the greatest integer function gives us:
\[ [x] = [5 + h] = 5 \]
because \(5 + h\) is still greater than \(5\) but less than \(6\).

### Step 4: Substitute into the Limit Expression
Now we can substitute this back into our limit expression:
\[ \lim_{x \to 5^+} (x - [x]) = \lim_{h \to 0^+} ((5 + h) - 5) \]

### Step 5: Simplify the Expression
This simplifies to:
\[ \lim_{h \to 0^+} h \]

### Step 6: Apply the Limit
As \(h\) approaches \(0\):
\[ \lim_{h \to 0^+} h = 0 \]

### Final Answer
Thus, the value of the limit is:
\[ \lim_{x \to 5^+} (x - [x]) = 0 \]

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