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A real-valued function f(x) is defined as: f(x) = tan²({x}) / ({x})² for x > 0; f(x) = 1 for x = 0; f(x) = sqrt({x}) * cot({x}) for x < 0, where {x} denotes the fractional part of x. Which of the following statements is INCORRECT?

1. lim x->0+ f(x) = 1
2. cot⁻¹( (lim x->0- f(x))²) = 1
3. tan⁻¹(lim x->0+ f(x)) = pi/4
4. lim x->0 f(x) exists

Correct answer: lim x->0 f(x) exists

Solution

For x->0+: {x} = x -> 0+, so f(x) = tan²(x)/x² -> 1. Left limit: for x->0-, {x} -> 1- (fractional part approaches 1 from below). So f(x) = sqrt({x}) * cot({x}) -> sqrt(1) * cot(1) = cot(1) ≈ 0.642. Since lim x->0+ f(x) = 1 ≠ cot(1) = lim x->0- f(x), the two-sided limit does NOT exist. So statement D ('lim x->0 f(x) exists') is INCORRECT. Checking A: lim x->0+ = 1, correct. Checking B: (lim x->0-)² = cot²(1); cot⁻¹(cot²(1)) is not obviously 1. Actually cot⁻¹(cot²(1)): cot(1) ≈ 0.642, cot²(1) ≈ 0.412, cot⁻¹(0.412) = arctan(1/0.412) ≈ arctan(2.43) ≈ 67.6 deg ≠ 1 rad. So B also appears incorrect. But the question asks which is INCORRECT among options — typically one answer. D is clearly wrong since the limit doesn't exist. D is the answer.

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