Limit Definition and 999 Threads

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...

Jupiter is huge. TrES-4 is 1.8 times the size. How big can planets actually get? is there a limiting factor? cheers.

##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...

$$\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n})$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi \sqrt{n^2+n}-n\pi)$$ $$=\lim_{n \rightarrow \infty} \sin^{2} (\pi (\sqrt{n^2+n}-n))$$ $$=\lim_{n \rightarrow \infty} \sin^{2}...

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...

Why is this not equivalent to 1 - inf^-inf, Or 1 - infinitesimal , Or 1 ? My answer was 1, which I told is incorrect.

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...

What is the difference of x->c^- and x->c^+?

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##...

6.2.25 DOY357 Evaluate $\displaystyle\lim_{x\to \infty}\dfrac{e^{3x}-e^{-3x}}{e^{3x}+e^{-3x}}$ sorry I'm clueless with this e stuff:(

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...