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Question:

Let f: R → R be a differentiable function satisfying f'(3) + f'(2) = 0. Then limx→0 (1 + f(3+x) - f(3))/(1 + f(2-x) - f(2))1/x is equal to

e

e - 1

e2

1

Solution:

Correct option is D. 1
limx→0 (1 + f(3+x) - f(3))/(1 + f(2-x) - f(2))1/x (1∞ form) ⇒ elimx→0 (f(3+x) - f(3) - f(2-x) + f(2))/x(1 + f(2-x) - f(2))
using L'Hopital ⇒ elimx→0 (f'(3+x) + f'(2-x))/ (1 + f(2-x) - f(2)) ⇒ ef'(3) + f'(2)/1 = 1