Manipulating Limits of Sequences

Calculus Level 4

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

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

How many of the following 6 statements are true:

AB,BC,CA,AC,BA,CB? 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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