Let `f: R to R` be a positive increasing function with `lim_(x to infty) (f(3x))/(f(x))=1`. Then `lim_(x to infty) (f(2x))/(f(x))` =

Options

A. 1
B. `2/3`
C. `3/2`
D. 3

Answer

To solve the problem, we start with the given information:

1. \( f: \mathbb{R} \to \mathbb{R} \) is a positive increasing function.
2. We know that \( \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \).

We need to find \( \lim_{x \to \infty} \frac{f(2x)}{f(x)} \).

### Step-by-step Solution:

**Step 1: Understanding the limit condition**

From the limit condition \( \lim_{x \to \infty} \frac{f(3x)}{f(x)} = 1 \), we can interpret that as \( f(3x) \) and \( f(x) \) becoming asymptotically equivalent as \( x \) approaches infinity. This suggests that the growth rates of \( f(3x) \) and \( f(x) \) are similar.

**Hint for Step 1:** Consider what it means for two functions to be asymptotically equivalent as \( x \to \infty \).

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**Step 2: Using the property of increasing functions**

Since \( f \) is an increasing function, we can say that for any \( c > 1 \):
\[ \lim_{x \to \infty} \frac{f(cx)}{f(x)} = 1 \]
This holds true for any constant \( c \) because as \( x \) becomes very large, the function's growth rate stabilizes.

**Hint for Step 2:** Think about how increasing functions behave when scaled by a constant factor.

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**Step 3: Applying the limit to our specific case**

Now, we can apply this property to our specific case where \( c = 2 \):
\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} = 1 \]

**Hint for Step 3:** Use the established property of limits for increasing functions to find the limit for \( c = 2 \).

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**Step 4: Conclusion**

Thus, we conclude that:
\[ \lim_{x \to \infty} \frac{f(2x)}{f(x)} = 1 \]

This means that as \( x \) approaches infinity, the ratio of \( f(2x) \) to \( f(x) \) approaches 1, indicating that \( f(2x) \) and \( f(x) \) are asymptotically equivalent.

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

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