Function Fails to Have Limit as x→∞, f(x)^2→4

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The discussion focuses on identifying a function f: R → R that does not have a limit as x approaches infinity, yet satisfies the condition that f(x)^2 approaches 4 as x approaches infinity. A proposed solution is the function f(x) = 1 if x is rational and f(x) = -1 if x is irrational. This function does not converge to a limit as x tends to infinity, while f(x)^2 equals 1, which does not meet the required condition of approaching 4.

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Juggler123
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I have been asked to give an example of a function form R to R that's fails to have limit as x tends to infinity but for which f(x)^2 tends to 4 as x tends to infinity.

I've had a go at this but the functions I can think of that don't have limits as x tends to infinity are the trigonemetric ones which doesn't help when looking at f(x)^2.

Can anyone help?
 
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Try f(x)= 1 if x is rational, -1, if x is irrational. That function has no limit as x tends to anything but f(x)2= 1.
 

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