Let`f(3)=4 and f'(3)=5`. Then `lim_(xrarr3) [f(x)]` (where [.] denotes the greatest integer function) is

Options

A. 3
B. 4
C. 5
D. non-existent

Explanation

To solve the problem, we need to evaluate the limit:

\[ \lim_{x \to 3} [f(x)] \]

where \( f(3) = 4 \) and \( f'(3) = 5 \). The notation \([.]\) denotes the greatest integer function (also known as the floor function).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
- We know that \( f(3) = 4 \), which means at \( x = 3 \), the function \( f(x) \) takes the value 4.
- The derivative \( f'(3) = 5 \) indicates that the function is increasing at \( x = 3 \).

2. **Finding the Limit from the Right**:
- We evaluate \( \lim_{x \to 3^+} f(x) \):
- Since \( f'(3) = 5 \), as \( x \) approaches 3 from the right, \( f(x) \) will be slightly greater than 4. Therefore, we can say:
\[ \lim_{x \to 3^+} f(x) = 4 + \epsilon \quad (\text{where } \epsilon \text{ is a small positive number}) \]
- Thus, the greatest integer function will yield:
\[ [\lim_{x \to 3^+} f(x)] = [4 + \epsilon] = 4 \]

3. **Finding the Limit from the Left**:
- Now, we evaluate \( \lim_{x \to 3^-} f(x) \):
- Since \( f'(3) = 5 \), as \( x \) approaches 3 from the left, \( f(x) \) will be slightly less than 4. Therefore, we can say:
\[ \lim_{x \to 3^-} f(x) = 4 - \epsilon \quad (\text{where } \epsilon \text{ is a small positive number}) \]
- Thus, the greatest integer function will yield:
\[ [\lim_{x \to 3^-} f(x)] = [4 - \epsilon] = 3 \]

4. **Conclusion**:
- We found that:
\[ \lim_{x \to 3^+} [f(x)] = 4 \quad \text{and} \quad \lim_{x \to 3^-} [f(x)] = 3 \]
- Since the left-hand limit and the right-hand limit are not equal, the overall limit does not exist:
\[ \lim_{x \to 3} [f(x)] \text{ does not exist.} \]

### Final Answer:
The limit \( \lim_{x \to 3} [f(x)] \) does not exist.

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