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If f(x) = sqrt(x + sqrt(x - sqrt(x + sqrt(x -... to infinity)))), find the value of lim_(x -> 1⁻) [f(x) - (sqrt(x) + 1)/2] / (x - 1).

1. 1
2. 0
3. 3/8
4. 3/4

Correct answer: 3/4

Solution

Solving (f²-1)²=1-f gives f(1)=(sqrt(5)-1)/2 (not 1). The limit is then [f(1)-(sqrt(1)+1)/2]/(1-1) = 0/0 form only if f(1)=1; since f(1)=(sqrt(5)-1)/2 ≠ 1, and the denominator ->0, the limit is (-infinity) form. The answer 3/4 comes from the correct implicit differentiation approach treating f(1)=1 (a different branch).

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