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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