Let f : R \rightarrow R be a positive increasing function with \( \lim_{x \to 0} \frac{f(3x)}{f(x)} = 1 \). Then \( \lim_{x \to 0} \frac{f(2x)}{f(x)} = \)
1
<span class="math-inline">\( \frac{2}{3} \)</span>
<span class="math-inline">\( \frac{3}{2} \)</span>
3
Step-by-Step Solution
Key Concept: A function with the property that lim_{x→0} f(ax)/f(x) = 1 for any positive constant a must have logarithmic behavior, meaning f(x) ≈ c·log|x| + d near x = 0. This allows us to evaluate similar limits involving different constants.
<p><strong>Step 1:</strong> Analyze what the condition lim_{x→0} f(3x)/f(x) = 1 tells us. Since f is positive and increasing, and this limit equals 1, the function f(x) cannot have a polynomial or exponential behavior near 0. The numerator and denominator grow/shrink at the same rate.</p><p><strong>Step 2:</strong> Consider the logarithmic form. If f(x) behaves like log|x| or any power less than any positive power near 0, then for any constants a, b > 0: lim_{x→0} f(ax)/f(bx) = 1. This is because f(ax)/f(x) = f(ax)/f(bx) · f(bx)/f(x), and if both individual ratios approach 1, their product does too.</p><p><strong>Step 3:</strong> Apply this principle systematically. We can write: lim_{x→0} f(2x)/f(x) = lim_{x→0} [f(2x)/f(3x)] · [f(3x)/f(x)]. Here, lim_{x→0} f(3x)/f(x) = 1 (given).</p><p><strong>Step 4:</strong> By the same reasoning applied to the ratio f(2x)/f(3x): since f is positive and increasing with the property that lim_{x→0} f(ax)/f(x) = 1 for any positive constant a, we must have lim_{x→0} f(2x)/f(3x) = 1.</p><p><strong>Step 5:</strong> Therefore: lim_{x→0} f(2x)/f(x) = 1 · 1 = 1.</p><p><strong>∴ Answer:</strong> 1</p>
Correct Answer: 1