If `lim_(xto3^(+))x/([x])`is ________________.

Answer

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

### Step-by-Step Solution

1. **Understanding the Limit**:
We need to evaluate the limit as \( x \) approaches 3 from the right, denoted as \( 3^+ \). This means we are considering values of \( x \) that are slightly greater than 3.

2. **Expressing the Limit**:
We can rewrite the limit as:
\[ \lim_{x \to 3^+} \frac{x}{[x]} \]

3. **Substituting Values**:
As \( x \) approaches 3 from the right, \( x \) can be expressed as \( 3 + h \), where \( h \) is a small positive number (i.e., \( h \to 0^+ \)). Thus, we can rewrite the limit:
\[ \lim_{h \to 0^+} \frac{3 + h}{[3 + h]} \]

4. **Evaluating the Greatest Integer Function**:
For \( x = 3 + h \) (where \( h \) is a small positive number), the greatest integer function \([3 + h]\) will equal 3, because \( 3 + h \) is still less than 4. Therefore, we have:
\[ [3 + h] = 3 \]

5. **Substituting into the Limit**:
Now we can substitute this back into our limit:
\[ \lim_{h \to 0^+} \frac{3 + h}{3} \]

6. **Simplifying the Expression**:
We can simplify the expression:
\[ \frac{3 + h}{3} = 1 + \frac{h}{3} \]

7. **Taking the Limit**:
As \( h \) approaches 0, the term \( \frac{h}{3} \) approaches 0. Therefore, we have:
\[ \lim_{h \to 0^+} \left(1 + \frac{h}{3}\right) = 1 + 0 = 1 \]

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 3^+} \frac{x}{[x]} = 1 \]