Limit Definition and 999 Threads

Background about myself (Can be skipped) I'm a gifted math student in the 11th grade, I studied Differential Calculus 2 years ago and I did pretty well in it, unfortunately I'm not studying math a lot these days, but I've gained a lot of experiences in life that made me more mature than I was 2...

Is there a theoretical limit to the size of neutron stars? It seems likely neutron stars are not simply electrons orbiting a proton so what is their life cycle? Can they just evaporate slowly by neutron decay?

In expressing a limit as below ...$$\text{lim}_{ x \rightarrow 0+ } \frac{ F( c + h ) - F(c) }{h} = f(c) $$How does one get the $$x \rightarrow 0+$$ to appear under the text "lim" as in the following:Help will be appreciated ... Peter

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 3: Sequences and Series ... I need help in fully understanding the proof of Theorem 3.4.11 ... Theorem 3.4.11 and the start of its proof read as...

I am reading "Introduction to Real Analysis" (Fourth Edition) b Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 3: Sequences and Series ... I need help in fully understanding an example given by B&S in some introductory remarks on Limit Superior and Limit Inferior ...The...

Homework Statement http://i66.tinypic.com/aesd1u.png can someone explain to me how can i get the limit using riemann sum especially the starred part? i was so confused thanks! Homework Equations The Attempt at a Solution attempt at a solution in the picture

Evaluate: \[\lim_{x\rightarrow 0}\frac{\sin (\arctan x)-\tan (\arcsin x)}{\arcsin (\tan x)-\arctan (\sin x)}\]

Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...

Homework Statement Prove ##~\displaystyle \lim_{x \to \infty}\left(x\sin\left(\frac{\pi}{x}\right)\right)=1## Homework Equations $$\lim_{x \to 0} \frac{\sin\pi}{x}=0$$ The Attempt at a Solution If i could multiply ##~x\sin\left(\frac{\pi}{x}\right)~## with something that would cancel the sin...

Homework Statement Compute Limit as x--> infinity of (logx)(log(logx)) / x The Attempt at a Solution Graphically, I see that the answer is perhaps zero, but I am not sure how to approach this algebraically. I worked at this for a couple hours, trying L'Hospital's rule but that did not really...

In the "Introduction to Solid State Physics" by C. Kittel, there is a long wavelength limit in chapter 4 -Phonons I. When Ka << 1 we can expand cos Ka ≡ 1 - ½ (Ka)2 the dispersion relation will become ω2 = (C/M) K2 a2 Does anyone know what frequencies can allow this long wavelength limit to hold?

Homework Statement Problem: Let ##A## be an infinite subset of a ##T_1## space, and let ##x## be a limit point of ##A##. Prove that every open neighborhood of ##x## contains infinitely many points of ##A##. Homework EquationsThe Attempt at a Solution First note that if ##\mathcal{O}## is an...

Homework Statement Use the definition of the derivative to find dy/dx for ##~y=\sqrt{2x+3}## Homework Equations Derivative as a limit: $$y'=\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$The Attempt at a Solution $$\lim_{\Delta x\rightarrow 0}\frac{\sqrt{2(x+\Delta...

Layman question here, kindly provide layman-understandable answers. The problem of quantum gravity is often expressed as saying 'GR predicts a collapse into a genuine singularity, there is no known mechanism which would stop such a collapse', and 'QM has nothing to say about gravity, it can not...

Hi All, Currently on a distance learning HNC and I am not quite sure whether the question just wants me to answer 'yes' or give mathematical evidence. Part A answered, Part B not sure... Any help would be great! 2. The process for the production of an electrical device is suitable for...

Homework Statement Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5 Homework EquationsThe Attempt at a Solution The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series...

Homework Statement I'm trying to do this limit based on a previous thread ( https://www.physicsforums.com/threads/proving-n-x-n-e-x-integrated-from-0-to-infinity.641947/#_=_ ) I got up to the last part of thread where I need to find the limit of: limit as x approaches infinity of...

Homework Statement Does the delta-epsilon limit definition in reverse work for describing limits in monotonic functions? By reversed, one means for lim (x -> a) f(x) = L if for each δ there corresponds ε such that 0 < | x-a | < δ whenever | f(x) - L | < ε. Homework EquationsThe Attempt at...

Hi everyone. I'll be grateful if someone can help me with this problem. 1. Homework Statement I have a closed system composed of one particle. The maximal velocity that this particle can have is equal to Vc. Here we consider only 2D space: X and Y direction. The particle velocity is V (which...

My screenshots show an attempt at a solution and the given solution path from the book, but I can't seem to figure it out.

So far in my reading of SR, we explore various consequences of c being a constant in a vacuum and frame invariant, etc. At what point in a physics education do you learn why a universal speed limit is necessary at all? Is that the sort of thing that is revealed in an intro to GR course? Or is...

Just read an article about a discovery of the smallest/least massive star in the Milky Way galaxy. The star has 85 times the mass of Jupiter and is known as EBLM J0555-57Ab located about 600 light-years from Earth. The entire article here -...

I understand that the standard proof is a bit different from my own, but I want to know if my reasoning is valid. PROOF: Firstly, I assume that x is positive. I then consider p = inf{n∈ℕ : n>x} . In other words, I choose "p" to be the smallest natural number greater than x. If we choose n>p...

Let $A_n$ be the area of the regular polygon whose vertices are given by the $n$ roots of unity in the complex plane. Prove: \[\lim_{n \rightarrow \infty }A_n = \pi \]

Is it possible to learn to prove limits by the formal definition without doing a course of real analysis? I'm not talking about just following the model that the Calculus books give, what I want is to understand the why of all the steps in formally proving the limit, to understand the why to use...

Hi. I'm reading a solution to a problem concerning a gas of photons. In the solution, the energy of the gas is given as E=2\sum_{\vec{k}} \frac{\displaystyle \epsilon_{\vec{k}}}{\displaystyle \exp[\beta\epsilon_{\vec{k}}]+1} where \epsilon_{\vec{k}} is one photon's energy. It is said then...

Why is it the case that, in a semiclassical description of the Einstein-Hilbert action, the cosmological constant is small in Planck units? Why does this mean that $$\ell \gg G$$ for ##\Lambda = - 1/\ell^{2}##?

How did Einstein first contemplate the idea that the speed of light was constant in all frames or reference? Did he say "I wonder what would happen if we considered light speed to be constant" in some kind of thought experiment. Did the concept fall serendipitously from the results of...

Homework Statement Find the limit of the following sequence: ##L_2 = \lim_{n \rightarrow +\infty} \frac {\sum_{k=0}^n (2k - 1)^p}{n^{p+1}}## Homework Equations 3. The Attempt at a Solution [/B] Seeing that ##\lim_{n \rightarrow +\infty} n^{p+1} = + \infty ## i can apply the Stolz theorem. (Is...

Homework Statement Give an example to show that the given "definition" of limx→aƒ(x) = L is incorrect. Definition: For each 0<δ there is an 0<ε such that if 0< l x-a I < δ , then I ƒ(x) - L I < ε . Homework EquationsThe Attempt at a Solution I considered the piece-wise function: ƒ(x) = (0 if...

Problem: Evaluate lim(x->0) x cotx My attempt: lim(x->0) x cotx = lim(x->0) x cosx / sinx = lim(x->0) cosx * lim(x->0) x / sinx = 1 * lim(x->0) x / sinx = lim(x->0) x / sinx P.S. I know I must/can use L'Hopital's rule to evaluate indeterminate limits, but no matter how many times I derive...

Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?

Homework Statement Find the above limit Homework Equations ##\lim_{t\rightarrow 0} \dfrac{\exp(-A/t)}{t^{n/2}}## with A>0 The Attempt at a Solution I tried using the change of variable ##u=1/t^{n/2}##, and then next use LHopital rule, but i don't find the way..

by Ken Croswell New observations indicate that objects born with a mass just 6.7 per cent that of the Sun can shine for trillions of years rather than fizzle out as failed stars known as brown dwarfs. Link: New Scientist

Homework Statement Calculate the following limit: Homework EquationsThe Attempt at a Solution I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression (x+\pi) to u, but I wasn't very sucessful. To what kind of algebric device I could...

I am trying to find the following limit ##\displaystyle \lim_{x \to 0^{+}} \frac{\log (t)}{\sqrt{t}}##, however, I don't see how. Obviously the answer is negative infinity, but I don't see how to get that. L'hospital's rule doesn't seem to work.

So, I have to show that in the non-relativistic limit the lower two components of the positive energy solutions to the Dirac equation are smaller than the upper two components by a factor of ##\beta##. I started with the spinor $$\psi = \begin{pmatrix} \phi \\ \frac {\vec \sigma \cdot \vec p}...

Homework Statement I'm in need of some help to be able to determine the supremum and infimum of the following sets:A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\}B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\}C = \left\{ {m\over \vert m\vert +n} \mid m \in...

I read a paper and i can't understand how can we make the conversion (that attached in photo) i mean how i get formula (5) from previous one

Hello all, I need some guidance in solving these limits: \[\lim_{x\rightarrow \infty }x\cdot sin(x)\] \[\lim_{x\rightarrow 0 }\frac{sin(x)}{\sqrt{x}}\] \[\lim_{x\rightarrow \infty }\frac{sin(x)}{x}\] I guess that the second and third ones are somehow related to \[\lim_{x\rightarrow 0...

Homework Statement I've begun going through Boas' Math Methods in the Physical Sciences and am stuck on problem 1.15.25. The problem is to evaluate ## \lim_{x\to \infty } x^n e^{-x} ## By using the Maclaurin expansion for ##e^{x}##. Homework Equations We know the Maclaurin expansion for the...

Homework Statement Calculate limit as x approaches infinity of (e^x - x^3) Homework Equations ln e^x = x e^(ln x) = x The Attempt at a Solution I tried substituting x = ln e^x and got (e^x - (ln e^x)^3). I'm pretty much lost and this is my only attempt so far. I'm thinking that this is an...

Hey! :o I am looking at the following exercise: Let $m_1\neq m_2$ be constants and $y$ the solution of the initial value problem $$y''-(m_1+m_2)y'+m_1m_2y=0\ \ \ \ y(0)=0 \ \ y'(0)=1$$ We consider $y$ as a function not only of $x$ but also of $m_1$ and $m_2$. With constant $m_2$ find (if...

Homework Statement I am wanting to show that ##lim_{z\to\infty} f(z)=c## does not exist for ##c \in C##, ##C## the complex plane, where ##f## is non-constant periodic meromorphic function. (elliptic) Homework EquationsThe Attempt at a Solution So I want to proove this is not true ...

I am doing a panel study with multiple linear regression. When I want to make sure that the residuals are normally distributed, as is a requirement for the regression model, can I assume so due the Central limit theorem (given the size is sufficient)? Or does it not apply when there is a time...

I'm stuck on this problem: find $$\lim_{{x}\to{0}} \frac{e^{x}-1}{\sin x}$$ Since no L'Hospital's Rule is allowed. I wonder if i can make use of the idea of punctured neighborhood(the current topic we learned in class) . Not sure how to set it up.

{\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N}\sum_{n=N+1}^{\infty}an is also converage proof that \lim_{N\rightarrow\infty}(\sum_{n=N+1}^{\infty}an)=0 {\displaystyle \sum_{n=1}^{\infty}a_{n}} is converage, For N\in \mathbb{N} \sum_{n=N+1}^{\infty}an is...

Homework Statement Homework Equations this is the entire syllabus of the test from which i got this question. Kindly mention if this question is out of syllabus[/B] The Attempt at a Solution I don't have the idea to approach this question. In normal limits, i know that graphically it is...

The definite integral of a function ##f(x)## from ##a## to ##b## as the limit of a sum is: $$\int_a^bf(x)dx=\lim_{h\rightarrow 0}h(f(a)+f(a+h)+.. ..+f(a+(n-2)h)+f(a+(n-1)h))$$ where ##h=\frac{b-a}{n}##. So, replacing ##h## with ##\frac{b-a}{n}## gives: $$\lim_{n\rightarrow...

A delightful video here A lecturer derives special relativity in a world of bats and echo location using a bat (sound) clock. All communication is limited by the speed of sound. (so a parallel of our world and light speed communication) He ends up observing that in no cases for any observer...