Let `f :RtoR` be a positive, increasing function with
`lim_(xtooo) (f(3x))/(f(x))=1`. Then `lim_(xtooo) (f(2x))/(f(x))` is equal to

Explanation

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

\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} \]

Given that:

\[ \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \]

### Step-by-step Solution:

1. **Understanding the properties of the function**:
- We know that \( f \) is a positive and increasing function. This means that for any \( x \), \( f(2x) > f(x) \) and \( f(3x) > f(2x) \).

2. **Setting up inequalities**:
- Since \( f \) is increasing, we can establish the following inequalities:
\[ f(x) < f(2x) < f(3x) \]
- Dividing through by \( f(x) \) (which is positive), we have:
\[ 1 < \frac{f(2x)}{f(x)} < \frac{f(3x)}{f(x)} \]

3. **Applying the limit**:
- Taking the limit as \( x \to \infty \):
\[ \lim_{x \to \infty} 1 < \lim_{x \to \infty} \frac{f(2x)}{f(x)} < \lim_{x \to \infty} \frac{f(3x)}{f(x)} \]
- We know from the problem statement that:
\[ \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \]
- Therefore, we can rewrite our limit expression:
\[ 1 < \lim_{x \to \infty} \frac{f(2x)}{f(x)} < 1 \]

4. **Concluding the limit**:
- The only value that satisfies \( 1 < L < 1 \) is \( L = 1 \).
- Thus, we conclude:
\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} = 1 \]

### Final Answer:
\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} = 1 \]