Limit Definition and 999 Threads

Homework Statement Let ##E'## be the set of all limit points of a set ##E##. Prove that ##E'## is closed. Prove that ##E## and ##\bar E = E \cup E'## have the same limit points. Do ##E## and ##E'## always have the same limit points? Homework Equations Theorem: (i) ##\bar E## is closed (ii)...

I'm having a difficult time researching the answer to my question about the speed of light. Now obviously it is a speed not only reserved for light but also all other massless particles/waves. It's obviously a constant property of our Spacetime since we can manipulate th speeds of different...

Am using Spivak and he defines limit of a function f 1. As it approaches a point a. 2.As it approaches infinity. He also defines limit f(x)=∞ x->a But though in solving exercises, we can see that all the three definitions are consistent with each other, I am not...

An interesting question has been posted by Brilliant.org. What is $\displaystyle \begin{align*} \lim_{x \to 0} \frac{x}{x!} \end{align*}$?My intuition tells me that the limit does not exist. My reasons for this are: 1. A limit can only exist if its left hand and right hand limits exist and...

The analysis of the distribution of spins for a paramagnetic solid in a B field shows that the probability of a dipole being aligned/anti-aligned with the B field ##\to 0.5## as ##T \to \infty##. The intuitive justifications that I've read say that this is "expected" as thermal motion tends to...

Homework Statement Evaluate the limit ##\lim_{x\to0} \frac{1}{x}(\frac{1}{tanx}-\frac{1}{x}) ## using Taylor's formula. (Hint: ##\frac{1}{1+c}=\frac{1-c^2+c^2}{1+c} ## may be useful) The Attempt at a Solution I began by substituting ##tanx## with ##x+\frac{x^3}{3}+x^3ε(x)##, where ε tends to...

Linear actuator spec.: 400mm stroke, 6000N, 2 limit switches, 5mm/sec, 12V DC supply, BIBUS IP65 casing. One of the limit switches of my LA (Linear Actuator) seems to have burned out. This happened because I accidentally plugged the LA leads into the AC mains socket (my LA is rated to run at...

Hello, I am struggling to understand a simple question on limits. I have watched a video trying to explain the theory and even have the answer right in front of me but I still don't understand. Could somebody please explain the steps in detail for me just for the first question as I'm hoping...

Homework Statement If ##a, b \in \{1,2,3,4,5,6\}##, then number of ordered pairs of ##(a,b)## such that ##\lim_{x\to0}{\left(\dfrac{a^x + b^x}{2}\right)}^{\frac{2}{x}} = 6## is Homework EquationsThe Attempt at a Solution So, this is a typical exponential limit...

Homework Statement a.) Let ##f,g:ℝ→ℝ## such that ##g(x)=sin x## and ##f(x)= \left\{ \begin{array}{ll} x^2, x∈ℚ \\ 0 , x∈ℝ\setminusℚ \\ \end{array} \right. ##. Calculate ##\lim_{x \rightarrow 0} \frac{f(x)}{g(x)}##. b.) Why l'Hospital rule cannot be applied here?The Attempt at a...

Homework Statement Calculate ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}## 'rigoriously'. The Attempt at a Solution What's the best approach? L'Hospitals rule? ##\lim_{x \rightarrow 0} \frac{arctanx}{arcsinx}=\lim_{x \rightarrow 0} \frac{\sqrt{1-x^2}}{x^2+1} =1##

If I am asked to show that the tt-component of the Einstein equation for the static metric ##ds^2 = (1-2\phi(r)) dt^2 - (1+2\phi(r)) dr^2 - r^2(d\theta^2 + sin^2(\theta) d\phi^2)##, where ##|\phi(r)| \ll1## reduces to the Newton's equation, what exactly am I supposed to prove?

Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states: (roughly translated from German) Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...

Homework Statement ##f(x)=12x^2-5## The value of ##\lim_{h\rightarrow 0}\frac{f(x+2h)-f(x-3h)}{6h}## is ... A. 8x B. 10x C. 12x D.18x E. 24x Homework Equations ##f'(x) = \lim_{h->0}\frac{f(x+h)-f(x))}{h}## The Attempt at a Solution [/B] Looking at the problem question, it seems that it's...

For the function given below find a formula for the Riemann sum obtained by dividing the interval [1,5] into n equal subintervals and using the right-hand endpoint for each c subscript k. Then take a limit of thissum as n-> infinite to calculate the area under the curve over [1,5]. Below you...

I've calculated the eigenstates of the Hubbard Hamiltonian for two fermions. The ground state is (U2 - (U2 + 16t2)1/2)/2 For U = infty, I get 0. For U >> t, I should get the exchange energy J = -4t2/U How do I get from the ground state equation to J?

I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...

Hi I was wondering how you get this when taking the limit of T going to 0 From this expression of S: Please help I don't see how ln infinity goes to uB/KbT (used u to represent the greek letter. And how does the other expression of sinh and cosh approach 1?

... literature review that we are thinking of attempting to publish? The review will be around 20,000 words and currently I have around 16000 words and 230 references, which seems quite a lot, is there a limit to how many references are present in a published review? This is not a thesis, but...

Homework Statement find the limit of arccoshx - ln x as x -> infinity Homework Equations ##arccosh x = \ln (x +\sqrt[]{x^2-1} )## The Attempt at a Solution ## \lim_{x \to \infty }(\ln (x + \sqrt{x^2-1} ) - \ln (x)) = \lim_{x \to \infty} \ln (\frac{x+\sqrt{x^2-1}}{x}) \ln (1 + \lim_{x \to...

I have heard that identical distinguishable classical particles having different ''statistics''.It is the limit of quantum case.Then we mix many parts(cells) of identical gases, the total entropy increases.I can not derive this limit from quantum particles to classical particles(please help...

Sunlight hits our planet, for example, and reflects light outward back into space. Hence why photos can be taken of Earth from outer-space. But if we disregard technological limits to optics, etc., then in theory how much information does this reflected light contain? Is it rich enough, for...

Homework Statement I want to find conditions over A,B,C,D to observe a stable limit cycle in the following system: \frac{dx}{dt} = x \; (A-B y) \hspace{1cm} \frac{dy}{dt} = -y \; (C-D x) Homework Equations Setting dx/dt=0 and dy/dt=0 you can find that (C/D , A/B) is a fixpoint. The Attempt...

Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online $$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{...

Homework Statement Let (an)n≥1 be a sequence with a1≥0 and an+1=an(an+4), n≥1. Compute limn→∞ (an)(1/(2n)). Homework Equations a1≥0 an+1=an(an+4), n≥1 L = limn→∞ (an)(1/(2n)) The Attempt at a Solution Firstly, I had tried to see if an can be expressed only in terms of a1, but I couldn't get...

Originally from the statistics forum but am told this is more of a calculus question. I flip 10 coins, if any of the coins land on tails, all of the coins split into 10 new coins and I flip them all again. I keep doing this until a round where every single coin lands on heads. Can I expect to...

Homework Statement A wooden cube with the length of 10 cm is and a density of 700 kg/m^3 is floating on water A)Find the submerged parts length of the cube B) find the maximum added mass to the block before it becomes totally submerged Homework Equations FB=density*volume*gravity FB=Fg...

Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...

A waiter believes the distribution of his tips has a model that is slightly skewed to the left, with a mean of $\$8.20$ and a standard deviation of ​$\$5.60$. He usually waits on about 60 parties over a weekend of work. ​a) Estimate the probability that he will earn at least ​$\$600$. ​b) How...

I've attached the equation for the transmission coefficient of a particle going through a potential barrier and E < V. I was simply wondering in the limit V --> E, why does T --> 0 (i.e. the V-E term --> 0 and thus the denominator would approach infinity, making T --> 0)? Shouldn't it be...

Homework Statement If f(a) = 0 and f'(a) = 6 find lim h -> 0 (f(a+h)/2h). Homework Equations lim h ->0 (f(a+h)-f(a))/h The Attempt at a Solution I found the ratio between the two equations. (f(a+h)-f(a))/h / (f(a+h)/2h) I found this to be 2. Is this step possible or can you not take the ratio...

I know that the limit for the spherical bessel function of the first kind when $x<<1$ is: j_{n}(x<<1)=\frac{x^n}{(2n+1)!} I can see this from the formula for $j_{n}(x)$ (taken from wolfram's webpage): j_{n}(x)=2^{n}x^{n}\sum_{k=0}^{\infty}\frac{(-1)^{n}(k+n)!}{k!(2k+2n+1)!}x^{2k} and...

Piran et al., "Cosmic explosions, life in the Universe and the Cosmological Constant," http://arxiv.org/abs/1508.01034 I thought this was interesting. If I'm understanding correctly, the idea is that satellite galaxies such as the Magellanic Clouds have low metallicity, which causes them to...

I have a limit problem. This is the problem: \lim_{x \to 2} \frac{\tan (2 - \sqrt{2x})}{x^2 - 2x} The solution is \lim_{x \to 2} \frac{\tan (2 - \sqrt{2x})}{x^2 - 2x} = \lim_{x - 2 \to 0} \frac{\tan [(2 - \sqrt{2x}) × \frac{2 + \sqrt{2x}}{2 + \sqrt{2x}} ]}{x(x - 2)} = \lim_{x - 2 \to 0}...

How to write limit like this in latex Instead of this

Hi everyone, So we were writting our math test today and I am not completely sure about one concept. For the sake of simplicity let's say that f(x)=x2 and let's say we were asked to find, lim f(x) as x--->infinity = ? is the correct answer here undefined or infinity. Thanks for the help

Why does the following Lagrangian not have the correct non-relativistic limit? It is correct except for the derivative of proper time with respect time. But that factor goes to 1 so why is the expression wrong? ## L = -(\frac{1}{2}mu^{\mu}u_{\mu} + qu^{\mu}A_{\mu})\frac{d\tau}{dt} ##

Homework Statement Prove that lim (x,y,z)→(0,0,0) 2xz/(x²+y²+z²) = 0 Homework Equations My teacher wants me to show this using epsilon delta, so 0<√(x²+y²+z²)<∂ ⇒ |f(x,y,z) - 0| < ε The Attempt at a Solution The limit does not exist apparently.. when you approach the limit along different...

When we integrate $f'(x)$ we get $f(x)$ and say we integrate from x = a to b in the output we write f(x) within square brackets and limit on the right. how do I write in latex thanks in advance.

Homework Statement For what value of the constants a and b such that the following limit exists? lim {(ax+|x+1|)|x+b-2|}/|x+1| x->-1 help me ,thx Homework EquationsThe Attempt at a Solution first, I know that I should cancel the absolute value at denominator of x+1. but i don't how to...

$$\lim_{{x}\to{0}}\frac{\cos\left({3x}\right)-1}{{x}^{2}}$$ $$\frac{f'}{g'} =-\frac{3\sin\left({3x}\right)}{2x} =-\frac{9}{2}\cdot\frac{\sin\left({3x}\right)}{3x}$$ $x\to 0$ is $-\frac{9}{2}$ Just seeing if this is correct or better way to do it

Homework Statement Find \lim_{x \to 0}\frac{ln(1+x^2)}{1-cos(x)} by using series representations. Check using L'Hospitals rule. Homework Equations Taylor polynomial at x=0: \sum_{k=0}^{\infty}\frac{f^{k}(0)}{k!}(x)^{k} = f(0) + f'(0)(x) + f''(0)x^{2} +... The Attempt at a Solution Using...

Homework Statement Assume f:(a,b)→ℝ is differentiable on (a,b) and that |f'(x)| < 1 for all x in (a,b). Let an be a sequence in (a,b) so that an→a. Show that the limit as n goes to infinity of f(an) exists. Homework Equations We've learned about the mean value theorem, and all of that fun...

Homework Statement [/B] Find the limit as ##n \to \infty ## of ##U_n(a) =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & a/n \\ 0 & -a/n & 1 \end{pmatrix}^n##, for any real ##a##. Homework EquationsThe Attempt at a Solution I find ##U =\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos a & \sin a \\ 0 & -\sin a &...

Homework Statement Hi, I'm currently studding a module on PLC and have a question on "what will happen if you do this ...". Homework Equations I don't have a problem with explaining operation of each rungs of the ladder diagram, I need some help on explaining the function of the reverse relay...

Hi, I'm currently studding a module on PLC and have a question on "what will happen if you do this ...". As I don't have a problem with explaining operation of each rungs of the ladder diagram I need some help on explaining the function of the reverse relay and travel limit switches. Can anyone...

In the system of units where c=1, the Lorentz transformations are as follows: ## x'=\gamma (x-vt) \\ t'=\gamma (t-vx) ## In the limit ## v \ll 1 ##, we have ## \gamma \approx 1+\frac 1 2 v^2 ##, so we have, in this limit: ## x' \approx (1+\frac 1 2 v^2)(x-vt)=x-vt+\frac 1 2 v^2 x-\frac 1 2...

I don't understand why the limit of x1/loga(x) as x approaches infinity is a, where a can be any constant for the base. Why isn't it 1? The base (x), approaches infinity, while the exponent approaches 0 (1/infinity), so it should be (infinity)0 = 1.

what value of the constants a and b if the following limit exists lim (ax + |x + 1|)|x + b − 2| |x + 1| x→−1 |x|= x for x≥ 0 and |x|= -x for x<0 |x+1|= x+1 for x≥ -1 and |x+1|= -(x+1) for x<-1 I don't know how to determine |x + b − 2| is positive or negative. i know that if limit...

Lim as x approaches 4 of 1/x = 1/4 Given epsilon > 0, come up with a delta, d?Limits have been introduced. So far my instructor has had us make tables to see what value x was approaching. Although I don't understand exactly how limits are EVALUATED (different from looking at a chart & saying...