In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (i.e., eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit.
The limit inferior of a sequence
x
n
{\displaystyle x_{n}}
is denoted by
lim inf
n
→
∞
x
n
or
lim
_
n
→
∞
x
n
.
{\displaystyle \liminf _{n\to \infty }x_{n}\quad {\text{or}}\quad \varliminf _{n\to \infty }x_{n}.}
The limit superior of a sequence
Was curious at the upper limit for neutron stars,
found this article stating one was found at around 700 / s
https://www.newscientist.com/article/dn8576-fast-spinning-neutron-star-smashes-speed-limit/
did not see the size, the article is behind a paywall, but it would have taken a radius of...
On one side, if I have any finite value of s = the side of the original triangle of the Koch snowflake iteration, then the perimeter is infinite, so intuitively
On the other hand, if I looked at the end result first and considered how it got there, then intuitively
(Obviously at n=infinity and...
In https://arxiv.org/pdf/1709.07852.pdf, it is claimed in equation (1) and (2) that when we take non-relativistic limit, the following Lagrangian (the interaction part)
$$L=g \partial_{\mu} a \bar{\psi} \gamma^{\mu}\gamma^5\psi$$
will yield the following Hamiltonian
$$H=-g\vec{\nabla} a \cdot...
Discussion: Assume that we can make ##\big| [\sqrt{4n^2 +n} - 2n]- \frac{1}{4}\big| ## to fall down any given number. Given an arbitrarily small ##\varepsilon \gt 0##, we assume
$$
\big| [\sqrt{4n^2 +n} - 2n] - \frac{1}{4}\big| \lt \varepsilon $$
$$
\big| [\sqrt{4n^2 +n} - 2n]\big| \lt...
It occurred to me that I should ask this to people who passed the stage in which I’m right now, being unable to find anyone in my milieu (maybe because people around me have expertise in other fields than mathematics) I reckoned to come here.
Let’s see this sequence: ## s_n =...
Let ##(M_i)_{i\in I}## be a multiverse of models of ZFC. By that I mean:
Each ##M_i## is a well-founded model of ZFC.
##(I,\leq_I)## is a partially ordered set, and whenever ##i\leq_I j##, there is an embedding ##\tau^j_i:M_i\rightarrow M_j## such that the image of ##M_i## is a transitive...
Hello,
I would like to ask, why there cannot be detected cosmic rays with energies higher than ~ 10^20 eV, i.e. beyond the GZK limit?
Thanks a lot in advance for the answer.
If this is not the correct forum, perhaps someone would be so kind as to move it to a more appropriate one? Thanks.
The current trend in computer chip manufacturing is towards making transistors smaller and smaller, so more and more can be packed in a single chip.
This has a number of...
It is reported that todays issue of Nature Magazine includes an article reporting a correlation between typical total life time gene mutations and typical life span in a variety of species. I do not subscribe to that magazine and have not it or its abstract on the web.
But ... in an article...
At non-relativistic limit, m>>p so let p=0
At non-relativistic limit m>>w,
So factorise out m^2 from the square root to get:
m*sqrt(1+2w(n+1/2)/m)
Taylor expansion identity for sqrt(1+x) for small x gives:
E=m+w(n+1/2) but it should equal E=p^2/2m +w(n+1/2), so how does m transform into p^2/2m?
Point B is elastic limit and point C is yield point.
From this link: https://en.m.wikipedia.org/wiki/Yield_(engineering)#Definition
The definition given is:
Both seems to refer to same definition, it is the point where the elastic deformation ends and plastic deformation begins. But from...
Summary:: Good afternoon. I have more questions about the details of epsilon-delta proofs. Below is a simple, rational limit proof example with questions at the end. The scratch work and proof are a bit pedantic but I don't follow proofs very well which omit a lot of details, including scratch...
I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts.
The Bendixson criterion is a theorem that permits one to establish...
Consider the series below;
From my own calculations, i noted that this series can also be written as ##S_n##=##\dfrac {3}{8}##⋅##\dfrac {4^n}{3^n}##. If indeed that is the case then how do we find the limit of my series to realize the required solution of ##1## as indicated on the textbook? I...
Good afternoon. I have some questions about the details of epsilon-delta proofs. Below is a simple, non-linear limit proof example which will serve as an example of the questions I have. The questions are below the example and involve clarification and explanations of steps and details in the...
Depth-limited search can be terminated with two Conditions of failure:
Standard Failure: it indicates that the problem does not have any solutions.
Cutoff Failure Value: It defines no solution for the problem within a given depth limit...
Summary:: standard failure in depth limit search.
Depth-limited search can be terminated with two Conditions of failure:
Standard Failure: it indicates that the problem does not have any solutions.
Cutoff Failure Value: It defines no solution for the problem within a given depth limit...
When I look at a range of inputs around x=c and consider the corresponding range of outputs
If 0< |x-c| <δ -> |f(x)-L1|<ϵ1 and |g(x)-L2|<ϵ2 as we shrink the range of inputs the corresponding outputs f(x) and g(x) narrow on L1 and L2 respectively.
|f(x)-L1||g(x)-L2|<ϵ2ϵ1
The product of the...
I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$
Then,
$$
\lim_{\varepsilon \searrow...
Being a neophyte to physics, I try to visualize a light cone as it travels about.
I try to put myself in it and use my car to talk of it.
When I ride in my my car, I note that when I corner, one wheel will speed up as compared to the other side.
A light cone does the same, and given that the...
Does the square of the sequence also have a limit of 1. Does the square root also equal 1? I've been trying to find some counterexamples but I think the limit doesn't change under these operations?
Suppose f1,f2... is a sequence of functions from a set X to R. This is the set T={x in X: f1(x),... has a limit in R}. I am confused about what is the meaning of the condition in the set. Is the limit a function or a number value? Why?
Prove that each of the limits exists or does not exist.
1. ##\text{lim}_{x\rightarrow 2}(x^2-1)=3##
##\text{lim}_{x\rightarrow 2}(x^2-1)=3## if ##\forall \epsilon>0, \exists \delta ## such that ##|x-2|<\delta \Rightarrow |f(x)-3|<\epsilon##.
\begin{align}&|x^2-1|=|x+1||x-1|\leq \epsilon\\...
I think when the speed of light was measured (and predicted from Maxwell's equations) that the assumption was made that this speed was a cosmic speed limit
Suppose that the cosmic speed limit was higher than c (not infinite) and that perhaps another form of radiation traveled at that speed...
Hi, PF
In a Spanish math forum I got this proof of a right hand limit:
"For a generic ##\epsilon>0##, in case the inequality is met, we have the following: ##|x^{2/3}|<\epsilon\Rightarrow{|x|^{2/3}}\Rightarrow{|x|<\epsilon^{3/2}}##. Therein lies the condition. If ##x>0##, then ##|x|=x##...
I just started using the Big_Integer library that is a part of the 202X version of ADA.
It is repeatedly described as an "arbitrary precision library" that has user defined implementation.
I was under the impression that this library would be able to infinitely calculate numbers of any length...
c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious?
First of all I don't think it is obvious but here is an explanation of why the limits are the same.
##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...
Attempt:
Note we must have that
## f>0 ## and ## g>0 ## from some place
or
## f<0 ## and ## g<0 ## from some place
or
## g ,f ## have the same sign in ## [ 1, +\infty) ##.
Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0##
Case 1: ##\forall \epsilon: 0<\epsilon<1##
$$\implies \epsilon^2<\epsilon<1$$
$$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$
Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty##
$$\epsilon\leq\epsilon^2...
I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##.
Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...
From Wikipedia:
Which should be conceptually similar of what happen in the non-relativistic limit of the Dirac equations when you see that the solutions decouple.
Do you have any reference that I can look up where the derivation for the KG field is performed?
Thanks in advance!
The answer sheet states that the series converges by limit comparison test (the second way).
In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way)
Thank you!
I have opposite conclusions about ##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}##
Quote from my textbook:
"The limit ##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}## is not nearly so subtle. Since ##-x>0## as ##x\rightarrow{-\infty}##, we have ##\sqrt{x^2+x}-x>\sqrt{x^2+x}##, which grows arbitrarily...
If I set x = 1, I can cancel out y-1 and get limit = 1
Now if I approach from the x-axis the numerator will be smaller or bigger than the denominator, but how would you prove that that does not result in 1 when you reach (x,y) = (1,1)?
TL;DR: Textbook says limit does not exist, but I obviously...
[Moderator's note: Thread spun off from previous thread due to topic/forum change.]
Time dilation sounds really weird, can i assume it has a logical explanation?
The limit itself is pretty easy to calculate
##lim_{T->0} \ lim_{\mu->\epsilon_F} \ (e^{\frac{(\epsilon_F - \mu)}{kT}}+1)^{-1} = \frac{1}{2}##
But I'm very confused about changing ##\epsilon_\vec{k}## to ##\epsilon_F##. Why do we do this?
Hi
I was working on a physics problem and it was almost solved.
Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it.
I hope you can help me.
This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...
In Apostol’s Calculus (Pg. 130) they are proving that 1/(x^2) does not have a limit at 0. In the proof, I am unable to understand how they conclude from the fact that the value of f(x) when 0 < x < 1/(A+2) is greater than (A+2)^2 which is greater than A+2 that every neighborhood N(0) contains...
Problem: Prove that $$\lim_{x\to 0}|\frac{|x|} x=\text{Undefined}$$
The solution written by my prof. uses a special case where the tolerance of error ##\epsilon=1/2##. However, I want to proof it with generality, meaning that the tolerance is shrinking.
Below is my attempt at a solution...
The universal speed limit is c, and as a consequence light is confined to that limit. I was thinking about the time dilation in SR and was wondering if this is result of reaching speeds close to the speed of light or because of reaching speed close to c?
For example, let's say light could be...
Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that:
$$
\lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0
$$
where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...
This was the question,
The above solution is the one that I got originally by the question setters,
Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand),
As you can see that both these methods give different answers...
Hi,
It's not a homework problem. I was just doing it and couldn't find a way to change the integral limit from "x" to "t". I should end up with kinetic energy formula, (1/2)mv^2. I've assumed that what I've done is correct. Thank you!
Edit:
"E" is work done.
From elementary calculus it is known that
(lim x-->0) ((sin x)/x) = 1.
Is this result equivalent to (lim x-->0) sin x = x ?
If so, how is it proved? Many thanks for all guidance.
I am currently obsessed with futurism but I am terrified I will run out of novums to contemplate about. A novum is an idea like “FTL travel” or “Gene splicing”. I was wondering if their is any proof that their is an unlimited amount of ideas that humans can come up with. My uncle was reading...