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Let f: (0, infinity) -> R be differentiable with f'(x) = 2 - f(x)/x for all x > 0 and f(1) != 1. Which statement is true?
- lim x->0+ f'(1/x) = 1
- lim x->0+ x f(1/x) = 2
- lim x->0+ x² f'(x) = 0
- |f(x)| <= 2 for all x in (0, 2)
Correct answer: lim x->0+ f'(1/x) = 1
Solution
f(x) = x + C/x, so f'(t) = 1 - C/t²; with t = 1/x -> infinity as x -> 0+, f'(1/x) -> 1.
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