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Evaluate the limit lim_(x→ 2)(√(1+√(2+x)-√(3)))/(x-2).

1. (1)/(8√(3))
2. (1)/(4√(3))
3. 0
4. None of these

Correct answer: (1)/(8√(3))

Solution

Let g(x)=sqrt(1+sqrt(2+x)). Then g(2)=sqrt(1+2)=sqrt3 and g'(x) = 1/(2g) * 1/(2 sqrt(2+x)). At x=2: g' = 1/(2 sqrt3) * 1/(2*2) = 1/(8 sqrt3). So the limit = 1/(8 sqrt3) = sqrt(3)/24.

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