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Let y = f(x) be the solution of the differential equation x² cos(1/x) dy/dx - y sin(1/x) = -1 with y -> -1 as x -> infinity. Which statement is correct?

1. f(x) = sin(1/x) - cos(1/x)
2. lim x->infinity (-f(x))^x = 1/e
3. lim x->infinity (-f(x))^x = e
4. f'(x) + 2x f'(x) + x² f''(x) = -(2/x²) sin(1/x)

Correct answer: lim x->infinity (-f(x))^x = 1/e

Solution

Solving gives f(x) = sin(1/x) + cos(1/x) (so the first option's form is wrong by a sign), and then (-f(x))^x -> 1/e as x -> infinity.

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