- #1

AdrianZ

- 319

- 0

## 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.