1 billion bloom zipf

Let f(x) be a function on [0,∞) such that
  limx→∞f(x) = ∞.
  Then there exists a unique continuous function g(x) on [0,∞) with
  limx→∞g(x) = ∞.
Suppose f and g are two differentiable functions on [0,∞) which satisfy
limx→∞f(x) = ∞ implies limx→∞g(x) = ∞
Then limx→∞f(x)/g(x) = 0 for all x in [0,∞). 
If you want to show that these limits exist then you need to prove that they both converge to some limit as x approaches infinity. This can be done by showing that 
limx→∞(f(x)/g(x))2 = limx→∞(f(x)/x)2 = limx→∞(f(x)/x + g(x)/x)2 = limx→∞(f(x)/x - g(x)/x)2 = limx→∞(f(x)/x - g(x)/x)2 = limx→∞(f(x)/x - g(x)/x)2 = limx→∞(f(x)/x - g(x)/x)2 = limx→∞(f(x)/x - g(x)/x)2 = limx→∞f(x)/x - g(x)/x