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Let f: (0, ∞) → R be a differentiable function such that f'(x) = 2 - f(x)/x for all x ∈ (0, ∞) and f(1) ≠ 1. Then
- lim x→∞ f'(1/x) = 1
- lim x→∞ x f(1/x) = 2
- lim x→∞ x² f'(x) = 0
- f(x) ≤ 2 for all x ∈ (0, 2)
Correct answer: lim x→∞ f'(1/x) = 1
Solution
Substituting f'(x) = 2 - f(x)/x into the limit expression and analyzing the behavior as x approaches infinity, it is found that lim x→∞ f'(1/x) = 1.
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