Prove that the limit of [x]+[-x] at infinity doesn't exist.

[AdrianZ]

Homework Statement

The problem is to prove that the limit of [x]+[-x] at infinity does not exist.

The Attempt at a Solution

I used the argument that the function [x]+[-x] is equivalent to the function f such that it gives 0 for all integers and gives -1 otherwise. therefore because the function f oscillates between 0 and -1 at infinity the limit can not exist. then I realized that my argument might be not so convincing for the professor. I then tried to use the following logical argument: assuming the limit exists and is equal to L, then there exists a positive number N that for any epsilon greater than zero when x>N we can conclude: |f(x) - L| < epsilon. therefore If I show that this assumption leads to a contradiction I have solved the problem. I said let's take x to be greater than N, f(x) is either 0 or -1 by definition. if f(x) is 0 then |0-L| < epislon which says L is between -epsilon and +epsilon, now since epsilon is an arbitrary number, let's take it to be equal to L/2. then it says that -L/2 < L < L/2 which is false.
Now if f(x) is -1, then |-1-L|=|L+1|<epislon. which says L is between -epsilon-1 and +epislon-1. now since epsilon is again an arbitrary number, let's assume that epsilon is equal to L. then it tells us that -L-1<L<L-1 and there is no such L that satisfies this inequality. therefore in both cases we've shown that the limit does not exist.

am I right? Can I write it to the professor as the answer?
Thanks in advance.

 

[micromass]

Hi AdrianZ!

Homework Statement

The problem is to prove that the limit of [x]+[-x] at infinity does not exist.

The Attempt at a Solution

I used the argument that the function [x]+[-x] is equivalent to the function f such that it gives 0 for all integers and gives -1 otherwise. therefore because the function f oscillates between 0 and -1 at infinity the limit can not exist. then I realized that my argument might be not so convincing for the professor. I then tried to use the following logical argument: assuming the limit exists and is equal to L, then there exists a positive number N that for any epsilon greater than zero when x>N we can conclude: |f(x) - L| < epsilon. therefore If I show that this assumption leads to a contradiction I have solved the problem. I said let's take x to be greater than N, f(x) is either 0 or -1 by definition. if f(x) is 0 then |0-L| < epislon which says L is between -epsilon and +epsilon, now since epsilon is an arbitrary number, let's take it to be equal to L/2. then it says that -L/2 < L < L/2 which is false.
Now if f(x) is -1, then |-1-L|=|L+1|<epislon. which says L is between -epsilon-1 and +epislon-1. now since epsilon is again an arbitrary number, let's assume that epsilon is equal to L. then it tells us that -L-1<L<L-1 and there is no such L that satisfies this inequality. therefore in both cases we've shown that the limit does not exist.

am I right? Can I write it to the professor as the answer?
Thanks in advance.

Your argument does not work for L=0. Indeed, you took epsilon to be equal to L/2. But if L is 0, then you've taken epsilon=0, which is not allowed. So you'll have to do something different for L=0.

In any case, I found your argument a bit weird (not that's it's wrong, it's just uncommon). You've basically fixed x and then said that there is no corresponding L. A more structured argument takes the following approach (which I'll leave to you to complete):

Assume the limit L exists, there are three possibilities for L:

• L=0. But this can't be because...
• L=1. But this can't be because...
• L is not 0 or 1. But this can't be because...

 

[AdrianZ]

Yes You're right. my argument doesn't work for L<0 as well. I was just thinking about this a few seconds ago before you reply.

I found your argument convincing, I mean you've assumed that there are 3 possibilities which is a right thing to do. Now let me try to work with your suggested argument.

well, assuming L=0 then we should have |f(x) - 0| < epsilon which says |f(x)| < 0 must be true for every f(x) depending on the epsilon that I choose. am I right? now If f(x) = -1 then it says that |-1|<0 which is a contradiction.

the second assumption too is easy to prove. but the third one is a little bit tricky. could you give me a clue of how to prove the 3rd one?

my first argument can be corrected if I take epsilon to be equal to |L/2| and then prove L=0 separately. am I right?

 

[micromass]

Yes You're right. my argument doesn't work for L<0 as well. I was just thinking about this a few seconds ago before you reply.

I found your argument convincing, I mean you've assumed that there are 3 possibilities which is a right thing to do. Now let me try to work with your suggested argument.

well, assuming L=0 then we should have |f(x) - 0| < epsilon which says |f(x)| < 0

No, it says that |f(x)|<epsilon for each f(x). Now what happens if you choose epsilon=1/2?

my first argument can be corrected if I take epsilon to be equal to |L/2| and then prove L=0 separately. am I right?

Yes, I believe so.

 

[AdrianZ]

well, sorry for being so confused at the beginning. I modified my argument this way, It turned out to be very similar to what micromass suggested:

assuming that the limit exists and is equal to L but not equal to 0 or -1, f(x) must be either 0 or -1. if f(x)=0 then |0-L|<epsilon which means |L|<epsilon. Now take the epsilon to be |L|/2 and we have come to a contradiction.
Now if f(x)= -1 then we have |-1-L|=|L+1|<epsilon. assuming that epsilon is equal to |L+1| leads to |L+1|<|L+1| which is false.
Now, assume that L=0. it suffices to show that there is one value for f(x) which leads us to contradiction. take f(x) = -1. then we'll have: |-1 - 0| < epislon which says that for any epsilon less than 1 the argument will fail which is a contradiction.
Now, assume that L=-1. it suffices to show that there is one value for f(x) which leads us to contradiction. take f(x) = 0. then we'll have |0 - (-1)| < epsilon which says that epsilon is greater than 1 which means any epsilon less than one would fail which is a contradiction.

Now, Is this one a correct argument?

 

[micromass]

Yes, that sounds good!

 

[AdrianZ]

Thank you micromass for your help. I do appreciate it xP