Proving or disproving f(x) < g(x) implies lim f(x) < lim g(x)

[htam]

This is my question:

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Let \(\displaystyle I\) be an open interval, and let \(\displaystyle a\, \in\, I.\) Suppose \(\displaystyle \displaystyle \lim_{x \rightarrow a}\, f(x)\, =\, L\) and \(\displaystyle \displaystyle \lim_{x \rightarrow a}\, g(x)\, =\, M.\)

Prove or disprove the following:

. . . . . . .\(\displaystyle \large{f(x)\, <\, g(x), \forall x\, \in\, I\, \Longrightarrow\, L\, <\, M}\)

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I think this has to be disproven. But I can't think of an instance where f(x) < g(x) and L = M.

 

[Dr.Peterson]

This is my question:

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Let \(\displaystyle I\) be an open interval, and let \(\displaystyle a\, \in\, I.\) Suppose \(\displaystyle \displaystyle \lim_{x \rightarrow a}\, f(x)\, =\, L\) and \(\displaystyle \displaystyle \lim_{x \rightarrow a}\, g(x)\, =\, M.\)

Prove or disprove the following:

. . . . . . .\(\displaystyle \large{f(x)\, <\, g(x), \forall x\, \in\, I\, \Longrightarrow\, L\, <\, M}\)

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I think this has to be disproven. But I can't think of an instance where f(x) < g(x) and L = M.

I see no mention of the functions being continuous. Can you find a counterexample where at least one is discontinuous at a?