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Let f(x) = sqrt(x + sqrt(x - sqrt(x + sqrt(x - sqrt(x...))))). 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

The nested radical f(x) satisfies a functional equation. At x=1, f(1)=1 and sqrt((x+1)/2)|ₓ₌₁=1, so the limit is 0/0 form -> use L'Hopital's rule: limit = f'(1⁺) - d/dx[sqrt((x+1)/2)]|ₓ₌₁. d/dx[sqrt((x+1)/2)]|ₓ₌₁=1/(2*sqrt(1))*1/2=1/4. Finding f'(1) requires differentiating the functional equation. The alternating nested radical: f = sqrt(x + g) where g = sqrt(x - f). So f²=x+g, g²=x-f. From these: f²-g²=(x+g)-(x-f)=g+f, so (f-g)(f+g)=f+g -> f-g=1 (assuming f+g!=0). So g=f-1. Then f²=x+f-1 -> f²-f-(x-1)=0 -> f=(1+sqrt(1+4(x-1)))/2=(1+sqrt(4x-3))/2. At x=1: f=1. f'(x)=2/(2*sqrt(4x-3))=1/sqrt(4x-3). f'(1)=1. Limit = f'(1) - 1/4 = 1-1/4... that gives 3/4. But check sqrt((x+1)/2) derivative: d/dx=1/(2*2*sqrt((x+1)/2))|ₓ₌₁=1/(4*1)=1/4. Limit=1-1/4=3/4. Answer 3/4? But option 3/8 is also there. Let me recheck. f=(1+sqrt(4x-3))/2. f'=(1/2)*4/(2*sqrt(4x-3))=1/sqrt(4x-3). f'(1)=1. sqrt((x+1)/2)' at x=1: (1/2)*(1/2)*(1/sqrt((x+1)/2))|ₓ₌₁=(1/4)/sqrt(1)=1/4. Limit=1-1/4=3/4.

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