If `f: R rightarrow (0, oo)` is an increasing function such tha `lim_(xrarroo) frac{f(7x)}{f(x)}=1` then the value of `lim_(xrarroo) [frac{f(5x)}{f(x)}-1]` (where [.] denote the greatest integer function) is

Options

A. 1
B. 0
C. 2
D. 3

Answer

To solve the problem, we need to analyze the given function \( f \) and the limits provided.

### Step-by-step Solution:

1. **Understanding the Limit**: We have the limit condition:
\[ \lim_{x \to \infty} \frac{f(7x)}{f(x)} = 1 \]
This implies that as \( x \) becomes very large, the values of \( f(7x) \) and \( f(x) \) become very close to each other.

2. **Using the Increasing Property**: Since \( f \) is an increasing function, we know:
\[ f(x) < f(5x) < f(7x) \]
for large \( x \). This means \( f(5x) \) is bounded between \( f(x) \) and \( f(7x) \).

3. **Applying the Limit to \( f(5x) \)**: We can express the limit for \( f(5x) \) as follows:
\[ \frac{f(5x)}{f(x)} < \frac{f(7x)}{f(x)} \quad \text{and} \quad \frac{f(5x)}{f(x)} > 1 \]
As \( x \to \infty \), since \( \frac{f(7x)}{f(x)} \to 1 \), it follows that:
\[ \lim_{x \to \infty} \frac{f(5x)}{f(x)} = 1 \]

4. **Finding the Desired Limit**: We need to find:
\[ \lim_{x \to \infty} \left( \frac{f(5x)}{f(x)} - 1 \right) \]
Since we established that \( \lim_{x \to \infty} \frac{f(5x)}{f(x)} = 1 \), we can substitute this into our expression:
\[ \lim_{x \to \infty} \left( \frac{f(5x)}{f(x)} - 1 \right) = 1 - 1 = 0 \]

5. **Applying the Greatest Integer Function**: The problem asks for the value of the limit in the context of the greatest integer function:
\[ \lfloor 0 \rfloor = 0 \]

### Final Answer:
The value of \( \lim_{x \to \infty} \left( \frac{f(5x)}{f(x)} - 1 \right) \) is \( 0 \).