Infinity Definition and 988 Threads

Homework Statement Finding the value of the limit: $$\lim_{t\to +\infty} t+\frac{1-\sqrt{1+a^2t^2}}{a}$$ ##a## is just a costant The Attempt at a Solution At first sight I had thought that the limit was ##\infty## but then I realized that there is an indeterminate form ##\infty - \infty##. I...

Since every point on a circle has exactly one other point (opposite its diameter) whose tangent is parallel, can it be said (proven?) that a circle is composed of an even number of points? It's messing with my head to think of infinity as even. I realize one-to-one mappings in infinite sets...

Greg Bernhardt submitted a new PF Insights post Intro to Big Bang and Infinity Concepts Continue reading the Original PF Insights Post.

Homework Statement ##\lim x \to \infty \frac{\sqrt{x+1} - \sqrt{x}}{\sqrt{3x + 5} - \sqrt{3x + 1}}## Homework EquationsThe Attempt at a Solution ##\lim x \to \infty {\sqrt{x+1} - \sqrt{x}} * \lim x \to \infty \frac{1}{\sqrt{3x + 5} - \sqrt{3x + 1}}## ##\lim x \to \infty \frac{(x+1) -...

Homework Statement Verify the following assertions: a) ##x^2 + \sqrt{x} = O(x^2)## 2. Homework Equations If the limit as x approaches ##\infty## of ##\frac {f(x)}{g(x)}## exists (and is finite), then ##f(x) = O(g(x))##. The Attempt at a Solution Let ##\epsilon > 0##. We solve for ##\delta##...

1. Homework Statement order of zero of a modular form ? 2. Homework Equations 3. The Attempt at a Solution Apologies if this is a stupid question but I'm pretty confused. So, a modular form ##f(t) \in M_k ## is usually given by it's expansion about ##\infty## expressed in the variable...

Hi I'm confused about something from quantum mechanics but it concerns infinity and limits. For an infinite well the energy levels vary as n2 and for an harmonic oscillator the energy levels vary as n with n taking integer values in both cases with no upper bound. In both cases there are...

The function that passes through (8,3) and every time x increases by 1 unit, the fuction decreases by 7 units

Hi, I am trying to create an HUD display - basically create a virtual image at infinity and look at it. I have two plano-convex lenses; the distance between them is the sum of their focal lengths; my object is therefore imaged at infinity. Here is the schematics that I'm pretty much trying to...

We were shown the answer to this question as a worked example: A photon is emitted from a radius ##r_2## and travels radially inward to ##r_1## until it's reflected by a fixed mirror and travels back to ##r_2##. Calculate the time taken for the photon to travel in and back, according to a...

Homework Statement Prove that {##x \epsilon \mathbb{R} : x^2 \ge 1##} is "not" bounded below. EDIT: I Looked closely and realized there is a "not" that we all had to write in...sorry if you lost some time.. Homework Equations Defintion: We say a nonempty subset ##A## of ##\mathbb{R}## is...

Now I expect that most of you on this forum would be familiar with the equality between point nine reoccurring and one: 0.999...=1 If your not familiar please review https://en.wikipedia.org/wiki/0.999... Now this equality can be used to imply something else, which is rather heterodox...

Hi PF! I am trying to integrate functions over an infinite domain. One example is $$\int_0^\infty \frac{e^{-x}}{\sqrt{x}}\,dx$$ I know the substitution ##u = \sqrt{x}## reduces this problem to integrating ##\exp(-x^2)##, but if I want to integrate the function as is, how would I do this? I've...

I have read some of the other posts about this topic but am still left unsatisfied. Could just be me. :cool: Did the universe, one minute after the big bang, consist of a finite volume of spacetime? If so, then is it not logically inconsistent that the universe can possibly be infinite now...

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...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ... I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ... I need help with an aspect of the proof of...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume 1: Foundations and Elementary Real Analysis" ... ... I am at present focused on Part 1: Prologue: The Foundations of Analysis ... Chapter 1: The Axioms of Set Theory ... I need help with an aspect of the proof of...

I was unsure whether or not to post this question here or in the Nuclear physics sub-section, but it's a relatively simple question: Given that quantum tunneling exists, would it be possible to produce infinite energy via repeated nuclear fusion reactions? Now given the second law of...

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...

Say we have two functions; ##f\left(x\right)=\frac{1}{e^x-1}## and ##g\left(x\right)=\ln \left(\frac{1}{x}+1\right)##. Let us find the limit of both functions as x approaches infinity; ##\lim_{x \rightarrow \infty} {f(x)} = \frac{1}{e^\infty-1} = \frac{1}{\infty} = 0## Therefore as ##x...

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?

Q1. Why is the probability current ##j(x,t)=0## at ##x=\pm\infty##? (See first line of last paragraph below.) My attempt at explaining is as follows: For square-integrable functions, at ##x=\pm\infty##, ##\psi=0## and hence ##\psi^*=0##, while ##\frac{\partial\psi}{\partial x}## and hence...

hi, say you have an infinite distance. is half of that distance also infinite? or 1/10000000000000 of that distance. and so on and so on. i suspect it must be because if you say that half of infinity is for example 10000 km then infinty must be 20000 km, which it is not. so, if this...

Homework Statement evaluate ## \sum _{n=1}^{\infty }4^{\frac{1}{n}}-4^{\frac{1}{n+2}}## . https://holland.pk/uptow/i4/fc981b864d95a636c4f08b9deb209cd6.png Homework Equations telescoping series: sum = infinite lim (a1-a(n+1)) S=a/(1-r) The Attempt at a Solution as the latter function is of...

i showed that an 2 = a02 if n is even and 1 - ao2 if n is even how am i supposed to evaluate this at infinity

Homework Statement Homework EquationsThe Attempt at a Solution let y = lim x->0+ x^cos(1/x) lny = cos(1/x)*lnx = (x*cos(1/x)) * (lnx/x) x*cos(1/x) = 0 (sandwich theorem) lnx/x = 0 (l'hopital) so lny = 0 and y = 1 Is this correct?

Consider the following paragraph taken from page 15 of Thomas Hartman's lecture notes (http://www.hartmanhep.net/topics2015/) on Quantum Gravity: In an ordinary quantum field theory without gravity, in flat spacetime, there two types of physical observables that we most often talk about are...

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 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 ...

Hello all, You are probably familiar with the problem of the infinity hotel, a hotel in which there is an infinite number of rooms. Each room is filled with a guest, i.e. there are infinite number of people too. It is well known that two infinite sets have the same number of elements if there...

If there is no upper limit on t, can you find a t such that: e^{iat} = e^{ia_0}, e^{ibt} = e^{ib_0} and e^{ibct} = e^{ic_0} at the same time? No matter what a,b and c is, though given a != b , a!=c, b!=c and a!= 0, b!= 0, c!=0 Or maybe rather: at=a_0 +k_12\pi, bt=b_0 +k_22\pi and ct=c_0...

I just read Carlo Rovelli's new book about loop quantum gravity. https://www.amazon.com/dp/0241257964/?tag=pfamazon01-20 Its a great book and i really enjoyed it. However there is a claim in that seems a little odd to me. In LQC singularities are resolved and replaced by bounces. So the the...

This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite: \begin{equation*} \psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty}...

What is the differential of infinity?

lim 1/x as x->0 is infinity, but the function taking it to infinity is continuous, but for continuous functions f(a)= lim f(x) as x->a, so by defininition 1/0 is infinity, what is wrong with this logic?

Is infinity a imaginary number?

I have seen at many places that if ever matter travels more faster than light, it's relativistic mass will reach nearly infinity. Some says it's the inertia, so very high energy is required to accelerate. But since it is traveling with the velocity above 3×10^8 m/s, i believe that the high...

What is infinity times zero? Isn't it zero? I mean, infinity times zero is the same as zero an infinite amount of times, and adding zero infinitely would give zero because even though zero is always being added, this process is the same as nothing ever being added to zero.

In what may be an obvious observation in all accounts. I realized I did not have as good a grasp on infinity as I thought I had. For infinity to be infinite in must go on for ever(obviously, this I get), however, it must reach a point on where it begins to return to numbers it has already used...

Homework Statement $$\lim_{x\to\infty} \left(\frac{n^2+2n+1}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}$$ Homework Equations 3. The Attempt at a Solution [/B] I tried ##\lim_{x\to\infty} \left(\frac{n^2+2n+3-2}{n^2+2n+3}\right)^{\frac{2n^2}{n+1}}=## ##\lim_{x\to\infty}...

I'm facing a problem in my physics course which is accepting that infinity can be a reference point in both Electrostatics (calculating the voltage of a point) and Matter Properties (calculating the gravitational potential energy), how come we use a reference point which we don't know where it...

How is it possible that there is a infinite amount of density at a point? I understand how number can be infinite but how does something tangible like matter reach infinity?

Hello! I'm doing a school project, where I am writing about Infinity in Math and Physics. I've got the math part settled, but it's the physics part that has begun to bother me. One part of my task is to write about some of the mathematical expressions in physics, where we use infinity - but the...

What is the equation for a curve where x approaches infinity as y approaches infinity?

Homework Statement An intensity current I descends down the z-axis from z = \infty to z = 0, where it spreads out in an isotropic way on the plane z = 0. Compute the magnetic field. Homework Equations The only relevant equation I can think of is Ampere's Law, \oint_\gamma \vec{B} \cdot...

Hi, I am having trouble with these kind of questions where we have to use L'Hospital's Rule. I took the ln of the function to get the x out of the exponent, and then followed the Rule by taking the derivative of the top and bottom (using a shortcut we learned: lim x --> infty f(x)g(x) = lim x...

Hi, I have ## 120 \sum \limits_1^\infty (\sigma_{3}(n))^{2} ## , where ## \sigma_{3}(n) ## is a divisor function. And I want to show that this can be written as ##120\sum \limits_{k=1}^{n-1} \sigma_{3}(k) \sigma_{3}(n-k) ## I'm pretty stuck on ideas starting of to be honest, since the sum is...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with the...

Infinity is both a number and a concept. I asked my 10 year old niece what kind of number infinity might be and she said, "It's a composite number." But I want to think about weather infinity is a prime number? Clearly if you divide infinity by any number, you get infinity. Also if you divide...