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

In summary, when a function fails to have a limit as x approaches infinity, it means that the values of the function do not approach a specific number or value as x gets larger. This can be determined by examining the behavior of the function as x gets larger, such as looking for asymptotes or analyzing the end behavior of the function. The square of the function approaching 4 does not necessarily impact the existence of the limit, as the limit is dependent on the behavior of the function itself. It is possible for a function to have a limit as x approaches infinity and for its squared value to also approach 4, indicating that the limit exists and is equal to 2. Additionally, a function can have a limit at infinity but still fail
  • #1
Juggler123
83
0
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 dosen't help when looking at f(x)^2.

Can anyone help?
 
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  • #2
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.
 

1. What does it mean when a function fails to have a limit as x approaches infinity?

When a function fails to have a limit as x approaches infinity, it means that the values of the function do not approach a specific number or value as x gets larger. This could be due to the function becoming undefined or oscillating between different values at infinity.

2. How can you determine if a function fails to have a limit as x approaches infinity?

A function fails to have a limit as x approaches infinity if the limit does not exist or is equal to infinity. This can be determined by examining the behavior of the function as x gets larger, such as looking for asymptotes or analyzing the end behavior of the function.

3. What impact does the square of the function approaching 4 have on the limit?

If the square of the function approaches 4, it does not necessarily mean that the limit exists. The limit of a function is dependent on the behavior of the function itself, not just its squared value. So, even if the squared function approaches a specific value, the limit may still not exist.

4. Is it possible for a function to have a limit as x approaches infinity and its squared value to also approach 4?

Yes, it is possible for a function to have a limit as x approaches infinity and for its squared value to also approach 4. This would mean that the limit of the function exists and is equal to the square root of 4, which is 2.

5. Can a function have a limit at infinity but fail to have a limit as x approaches infinity?

Yes, a function can have a limit at infinity but fail to have a limit as x approaches infinity. This can occur if the function has a horizontal asymptote at infinity, where the limit exists but the values of the function do not approach a specific number as x gets larger.

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