Is the converse of the absolute value limit theorem true?

In summary, the conversation discusses the relationship between the limit of a function and the limit of its absolute value. It is proven that if the limit of a function is L, then the limit of its absolute value is abs(L). However, the converse is not always true, as shown by the counterexample of the map f:R-->R defined by f(x)= 1 if x is rational and =-1 if x is irrational. It is also discussed that for a sequence to converge to a limit, it must also converge to the absolute value of that limit.
  • #1
nitro
9
0
Hi all,
I'm wondering about this question

I can prove that if lim_{n->inf} (a) = L then lim_{n->inf}abs(a) = abs(L)
however.. is the converse true?
thx
 
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  • #2
Consider the map f:R-->R defined by

f(x)= 1 if x is rational and =-1 if x is irrational

Then |f| is identically 1 so it is everywhere continuous, while f is nowhere continuous by the density property of the rational and irrational numbers.

In particular, for instance, let x_n be a sequence converging to 0 that contains an infinity of rational numbers and an infinity of irrational numbers and set a_n :=f(x_n). Then |a_n|-->1 but a_n does not converge because it has a subsequence converging to 1 and another converging to -1.

However, if |a_n|-->L and a_n converges, then it must be that a_n-->±L because if a_n-->P, and if

|a_n-P|<epsilon,

as soon as n>N, then because of the triangle inequality

||a_n|-|P||<=|a_n-P|,

it follows that

||a_n|-|P||<epsilon

as soon as n>N also.

But this is the same as saying that |a_n|-->|P|. So L=|P|. So P=±L.
 
  • #3
Considering the constant sequnce -1 would also do the job. Its absolute value convreges to 1, so you can choose L=1 (of course you could also choose l=-1:smile:, but you don't have to.)
Then the sequnece does not converge to L...
 

1. What is the definition of the absolute value of a limit?

The absolute value of a limit is the distance between the limit and the value that the function approaches as the independent variable approaches a certain value. It is always a positive number, regardless of the direction in which the limit approaches the value.

2. How is the absolute value of a limit calculated?

The absolute value of a limit is calculated by taking the limit of the function and then taking the absolute value of that result. This means finding the distance between the limit and the value approached by the function.

3. What is the purpose of finding the absolute value of a limit?

The absolute value of a limit is useful in determining the behavior of a function as the independent variable approaches a certain value. It can help identify if the function has a limit at that point, and if so, what that limit is.

4. Can the absolute value of a limit be negative?

No, the absolute value of a limit is always a positive number. This is because it represents the distance between the limit and the value approached by the function, which cannot be negative.

5. How can we use the concept of absolute value of a limit in real-world applications?

The concept of absolute value of a limit is important in various fields of science and engineering, such as physics, chemistry, and economics. It can be used to analyze the behavior of systems and predict their outcomes, as well as to optimize processes and make informed decisions based on the behavior of a function.

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