Limit Superior/Inferior

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In summary, we discussed the definitions of limsup and liminf, and how they differ from the conventional limit. We also looked at an example function and saw that the limsup and liminf were equal to the limit at points where the function had an actual limit, but not necessarily at other points. We also clarified that limsup and liminf are the maximum and minimum of the limit points, respectively, and not the limit itself.
  • #1
kaosAD
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I have questions regarding this subject.

By definition, [tex]\limsup_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \sup_{n \geq k} f(x_n)[/tex] and [tex]\liminf_{k \to \infty} f(x_k) \equiv \lim_{k \to \infty} \inf_{n \geq k} f(x_n)[/tex]. Say a sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the left in the following example.

[tex]f(y) = \left\{
\begin{array}{ll}
y + 1 & \quad ,y > 0 \\
y & \quad ,y \leq 0
\end{array}
\right.[/tex]

Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) = f(0)[/tex].

Suppose we have another sequence [tex]\{x_k\}[/tex] converging to [tex]0[/tex] from the right. Then [tex]\limsup_{k \to \infty} f(x_k) = \liminf_{k \to \infty} f(x_k) > f(0)[/tex].

What is the difference between [tex]\limsup_{k \to \infty} f(x_k)[/tex] and [tex]\liminf_{k \to \infty} f(x_k)[/tex]? I don't see any difference.
 
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  • #2
I'm very confused as to what you're actually doing.

Anyways, your function has an actual limit at [itex]+\infty[/itex]: in particular,

[tex]\lim_{y \rightarrow +\infty} f(y) = +\infty[/tex]

so of course the lim sup and lim inf are going to be equal.


I suspect you meant to take a one sided limit at zero.. but again, the function has an actual limit there:

[tex]\lim_{y \rightarrow 0^+} f(y) = 1[/tex]

so again, the lim sup and lim inf are going to be equal.



Oh, I think I've figured out what you're trying to do.

What you're forgetting is that

[tex]\limsup_{y \rightarrow 0} f(y)[/tex]

is the supremum over all sequences that converge to zero and for which [itex]f(x_k)[/itex] converges. So just picking a particular sequence and saying that the lim sup equals the limit of that sequence is wrong.
 
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  • #3
Hurkyl said:
I'm very confused as to what you're actually doing.
Sorry for not explaining my problem clearer.

Hurkyl said:
Oh, I think I've figured out what you're trying to do.

What you're forgetting is that

[tex]\limsup_{y \rightarrow 0} f(y)[/tex]

is the supremum over all sequences that converge to zero and for which [itex]f(x_k)[/itex] converges. So just picking a particular sequence and saying that the lim sup equals the limit of that sequence is wrong.

That was what my problem was. Many thanks, Hurkyl! :)
Does this mean [itex]\lim_{k \to \infty} f(x_k)[/itex] is the limit over all sequences as well?

Also, using the given example, can I write [itex]0 \leq \lim_{x \to 0} f(x) \leq 1[/itex]? or that [itex]\lim_{x \to 0} f(x)[/itex] does not exist, so we cannot write down its range?
 
  • #4
You can't write that expression for the limit, because as you said, the limit is undefined.

You can say that all of the limit points are in that interval, though, because lim inf and lim sup are the minimum and maximum of the limit points.
 

What is the limit superior/inferior?

The limit superior/inferior is a concept in mathematical analysis that describes the behavior of a sequence of numbers as it approaches infinity or negative infinity. It is denoted by lim sup for limit superior and lim inf for limit inferior.

How is the limit superior/inferior calculated?

The limit superior/inferior is calculated by finding the highest/lowest limit points of a sequence. This is done by taking the supremum (least upper bound) or infimum (greatest lower bound) of the set of all possible accumulation points of the sequence.

What is the difference between limit superior and limit inferior?

The limit superior is the largest limit point of a sequence, while the limit inferior is the smallest limit point. In other words, the limit superior describes the upper bound of the sequence, while the limit inferior describes the lower bound.

What is the importance of limit superior/inferior in mathematics?

The concept of limit superior/inferior is important in mathematical analysis because it helps to determine the behavior of a sequence as it approaches infinity or negative infinity. It also allows for the comparison and classification of different types of sequences.

How is the limit superior/inferior used in real life?

The limit superior/inferior is used in various fields of science and engineering, such as physics, computer science, and economics. It is used to model and analyze various phenomena, such as stock market trends, population growth, and the behavior of physical systems.

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