Manipulating Limits of Sequences

Calculus Level 4

Consider the following statements about positive functions $$f(x)$$ and $$g(x)$$, whose limits to infinity exists:

A) $$\lim_{ x \rightarrow \infty} f(x) = \lim_{ x \rightarrow \infty} g(x)$$.
B) $$\lim_{ x \rightarrow \infty} f(x) - g(x) =0$$.
C) $$\lim_{ x \rightarrow \infty} \sqrt{ f(x)} = \lim_{ x \rightarrow \infty} \sqrt{ g(x) }$$.

How many of the following 6 statements are true:

$A \Rightarrow B, B \Rightarrow C, C \Rightarrow A, A \Rightarrow C, B \Rightarrow A, C \Rightarrow B ?$

This is related to the discussion AM=GM when infinite?

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