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
My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral.
My work is below:
If we consider the canonical ensemble then, after tracing over the corresponding exponential we get:
$$Z = \sum_{n=0}^\infty...
First i looked at the case of ## \epsilon = 0## and came to the conclusion, that this oscillator has a circular limit cycle in a phase space trajectory, when plotted with the axes x and ##\dot{x}##.
I have found that ##x_0^p (t) = a_1 \cos(t)## which implies that all other Fourier- coefficients...
For this problem,
The limiting position of R is (4,0). However, I am trying to solve this problem using a method that is different to the solutions. So far I have got,
##C_1: (x - 1)^2 + y^2 = 1##
##C_2: x^2 + y^2 = r^2##
To find the equation of PQ,
## P(0,r) ## and ##R(R,0) ##
## y =...
I think that the work is meant to be work done for instance in power stations. Or is it similar to work I do on a body when I lift it for example? But how can we then do that work on our Earth? I just need to understand the task, otherwise I want to solve it myself.
The problem involves...
For this problem,
The solution is,
However, I tried to solve this problem using my Graphics Calculator instead of completing the square. I got the zeros of ##x^2 - 2x - 4## to be ##x_1 = 3.236## and ##x_2 = -1.236##
Therefore ##x_1 ≥ 3.236## and ##x_2 ≥ -1.236##
Since ##x_1 > x_2## then...
This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
What is the limit of the function as x goes to -5 (e.g. in the graph below) if the domain of the function is only defined on the closed interval [-5,5]?
I realize that the right hand limit DOES exist and is equal to 3, but the left hand limit does not exist?
So does that mean that the overall...
For this problem,
The solution is,
However, when I tried finding the domain myself:
## { x | x - 1 ≥ \sqrt{5}} ## (Sorry, for some reason the brackets are not here)
##{ x | x - 1 ≥ -\sqrt{5}} ## and ## { x | x - 1 ≥ \sqrt{5}}##
##{x | x ≥ 1 -\sqrt{5} }## and ## { x | x ≥ \sqrt{5} + 1}##...
Starting from the Heisenberg equation of motion, we have
$$ih \frac{\partial p}{\partial t} = [p, H]$$
which simplifies to $$ih \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}$$
but this just results in ## \frac{\partial p}{\partial t} = -ih\frac{\partial V}{\partial x}## and...
Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt:
(i) General Riemann Sums
Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
Hi,
I had to calculate the entropy in a task of a lattice gas and derive a formula for the pressure from it and got the following
$$P=\frac{k_b T}{a_0}\Bigl[ \ln(\frac{L}{a_0}-N(n-1)-\ln(\frac{L}{a_0}-nN) \Bigr]$$
But now I am supposed to calculate the following limit
$$\lim\limits_{a_0...
Let ##c## be a complex number with ##|c| \neq 1##. Find $$\lim_{n\to \infty} \frac{1}{n}\sum_{\ell = 1}^n \frac{\sin(e^{2\pi i \ell/n})}{1-ce^{-2\pi i \ell/n}}$$
In Section 3.4 of Shapiro & Teukolsky (1983), a simple derivation, due to Landau, of the maximum mass limit for white dwarfs and neutron stars is given. I will briefly describe it here and then pose my question.
The basic method is to derive an expression for the total energy (excluding rest...
I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
This is probably a dumb question. I'm not a physicist and took basic physics a very long time ago.
If an object was in deep space, a long way away from gravitational fields and was subjected to a constant 1g acceleration in a straight line what prevents it from eventually exceeding light speed...
If we have two functions ##f(x)## such that ##\lim_{x \to \infty}f(x)=0## and ##g(x)=\sin x## for which ##\lim_{x \to \infty}g(x)## does not exist. Can you send me the Theorem and book where it is clearly written that
\lim_{x \to \infty}f(x)g(x)=0
I found that only for sequences, but it should...
I'm trying to compute ##\int_0^1 x^m \ln x \, \mathrm{d}x##. I'm wondering if the bit about the application of L'Hopital's rule was ok. Can anyone check?
Letting ##u = \ln x## and ##\mathrm{d}v = x^m##, we have ##\mathrm{d}u = \frac{1}{x}\mathrm{d}x ## and ##v = \frac{x^{m+1}}{m+1}##...
Right now, I am trying to prove this :
I tried to use this identity to solve it:
Then, the limit will become ##\frac {x}{e-e}##
However, the result is still ##\frac 0 0 ##
Could you please give me hints to solve this problem?
\lim_{x \rightarrow 1} \frac{x^2 - 1}{x-1}
For this, we first divide the numerator and denominator by (x-1) and we get
\lim_{x \rightarrow 1} (x+1)
Apparently, we can divide by (x-1) because x \neq 1, but then we plug in x = 1 and get 2 as the limit. Is x = 1 or x \neq 1? What exactly is...
I'm reading《Introducing Einstein's Relativity_ A Deeper Understanding Ed 2》on page 180,it says:
since we are interested in the Newtonian limit,we restrict our attention to the spatial part of the geodesic equation,i.e.when a=##\alpha####\quad ##,and we obtain,by using...
Hello everyone,
Our topology professor have introduced the standard topology of ##\mathbb{R}## as:
$$\tau=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left(x-\delta,x+\delta\right)\subset u\right\},$$
and the lower limit topology as...
Indeterminate forms are: ##\frac{0}{0}, \frac{\infty}{\infty} , \infty - \infty, 0 . \infty , 1^{\infty}, 0^{0}, \infty^{0}##
My answer:
4, 9, 15, 17, 20 are inderterminate forms
1. always has a fixed finite value, which is zero
2. ##0^{-\infty}=\frac{1}{0^{\infty}}=\frac{1}{0}=\infty## so it...
##i^\frac{1}{n}## has n roots. If one is not careful, the limit as ##n \to \infty## is 1. Simple proof: ##i=e^\frac{\pi i}{2}## or ##i^\frac{1}{n}=e^\frac{\pi i}{2n} \to e^0=1##.
This does not take into account the n roots, since ##i=e^{(\pi i)(2k+\frac{1}{2})}##.. Here ##\frac{k}{n} ## can...
I imagine ##f(x)## has horizontal asymptote at ##x=k##. Since the graph of ##f(x)## will be close to horizontal as ##x \rightarrow \infty##, the slope of the graph will be close to zero so ##\lim_{x \rightarrow \infty} f'(x) = \lim_{x \rightarrow \infty} f^{"} (x) = 0##
But how to put it in...
##f'(x_0)## is defined as:
$$f'(x_0)=\lim_{h \rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h}$$
or
$$f'(x_0)=\lim_{x \rightarrow x_0} \frac{f(x)-f(x_0)}{x-x_0}$$
I can imagine that as ##n \rightarrow \infty## the value of ##f(b_n)## and ##f(a_n)## will approach ##f(x_0)## so the value of the limit will...
I know $$\lim_{h\rightarrow 0} af(h)+bf(2h)−f(0)=0$$
$$a+b=1$$
But I don't know how to find the second equation involving a and b. I imagine I need to somehow obtain ##h## in numerator so I can cross out with ##h## in denominator but I don't have idea how to get ##h## in the numerator.
Thanks
Summary: Definition: If M is a set and p is a point, then p is a limit point of M if every open interval containing p contains a point of M different from p.
Prove: that if H and K are sets and p is a limit point of H ∪ K,then p is a limit point of H or p is a limit point of K
In this proof I...
I have a probability distribution of the following form:
$$\displaystyle f \left(t \right) \, = \, \frac{\lambda ~e^{-\frac{\lambda ~t }{k }}}{k }, \, 0 < t, \, 0< \lambda, \, k = 1, 2, 3, \dots$$
It seems that this distribution is a limiting case of another distribution. The question is what...
The original differential equation is:
My solution is below, where C and D are constants. I have verified that it satisfies the original DE.
When I apply the first boundary condition, I obtain that , but I'm unsure where to go from there to apply the second boundary condition. I know that I...
Let's say we're given a sequence ##(s_n)## such that ##\lim s_n = s##. We have to prove that all subsequences of it converges to the same limit ##s##. Here is the standard proof:
Given ##\epsilon \gt 0## there exists an ##N## such that
$$
k \gt N \implies |s_k - s| \lt \varepsilon$$
Consider...
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...