lim(x →∞) f(x) is equal to (A) -∞ (B) ∞

Question

Let f be function satisfying the condition  f(xy) = f(x)/y + f(y)/x ∀ x, y > 0. If f(x) is differentiable and f(1) = 1, then

lim_(x →∞) f(x) is equal to

(A) -∞

(B) ∞ 

(C) 0 

(D) 1

Answer 1

Answer is (C) 0

This is a linear differential equation. 

Solution is f(x)· x = ln x + c

Applying L’ Hospital’s rule,