Limit Definition and 999 Threads

...Or even based on logic? I understand that it is expected that there might be a Universal Speed Limit and that this seems with extremely high probability to coincide with the speed of em transmission in a vacuum. This is borne out by experimentation and observation. Are there any other...

I tried substituting x=cos2theta but it was of no use.I thought many ways but i could not make 0/0 form.So please help.

I wrote cos(pi(n^2+n)^(1/2)) as cot(pi(n^2+n)^(1/2))/cosec(pi(n^2+n)^(1/2)) and as we know cot(npi)=infinity and cosec(npi)=infinity , so i applied L'Hospital.After i differentiated i again got the same form but this time cosec/cot which is again infinity/infinity.But if i differentiate it i...

Hey! :o We consider the equation \begin{equation*}u_{k+1}=\begin{pmatrix}a & b \\ 1-a & 1-b\end{pmatrix}u_k \ \text{ with } \ u_0=\begin{pmatrix}1 \\1 \end{pmatrix}\end{equation*} For which values of $a$ and $b$ is the above equation a Markov process? Calculate $u_k$ as a function of $a,b$...

I've recently been reading about the 2-dimensional Ising model and its continuum limit from several sources, including https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture1.pdf https://webhome.weizmann.ac.il/home/fnfal/papers/Ising/lecture2.pdf As far as I understood it, the state...

Question in the title .

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Is it known how much momentum a photon possesses if it's wavelength is at the Planck length, and what happens if it's momentum is somehow increased from that?

Ans is 3. I know basic t-test but I have no clue to solve this question. Thanks.

Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...

I am working with a simulation which generates an arbitrary number ##n## of identical curves with different phases and calculates their (normalized) sum. As expected, the fluctuation depth of the curves decreases as we increase ##n##. Here is an example of my simulation (when ##n>1##, the...

Express the limit ##lim_{n\rightarrow\infty} \sum_{i=1}^n \frac2n\ (1+\frac {2i-1}{n})^\frac13## This is worked example but I would like to ask about the points I don't understand in the book. "We want to intepret the sum as a Riemann sum for ##f(x)=(1+x)^3## The factor ##\frac2n## suggests...

Let $m_n$ be the smallest value of the function: $$f_n(x)=\sum_{k=0}^{2n}x^k.$$ Show, that $m_n\to\frac{1}{2}$ as $n \to \infty$. Source: Nordic Math. Contest

Finding $$\lim_{n\rightarrow \infty}\sqrt{n}\int^{\frac{\pi}{4}}_{0}\cos^{2n-2}(z)dz$$

This is going to take a while to set up, so I apologize for that. This came up in the course of thinking about the Strong Law of Large Numbers. It's not homework. Suppose you have a doubly infinite sequence of random variables X_{i,n} that obey the following almost sure convergence relations...

I have been reading about the quantum effects that limit the Maxwell-Boltzmann Distribution under certain conditions which leads to the Bose-Einstein or Fermi-Dirac Distribution. I have difficulty grasping the reasons why these quantum-effects occur only at certain conditions and why exactly...

Hello, I'm having a problem with this system. Ignore the physics. I have the feeling it should be tremendously easy... but I can't figure it out. I don't know how to extract it from the pdf so I'll post just the these 2 pages. https://ufile.io/39ovq The equations are (1.14) and (1.15), the...

Homework Statement Let f be a function from R2 to R. Suppose that f (x, y) → 3 as (x, y) approaches (0,1) along every line of the form y = kx + 1. What can you say about the limit lim(x,y)→(0,1) f (x, y)? Check the box next to the correct statement. Homework Equations N/A The Attempt at a...

Admitted I know very little about QM, but I've been thinking about black holes and I wondered if there would be an upper limit to density of an object of the smallest size allowable if the particles are not being observed by anyone (since black holes are black)? I ignorantly wondered that...

First time poster here, thanks in advance! I have a project I'm working on, and I'm looking for a way to limit the rotational motion of a shaft inside a cylinder. The cylinder is fixed, and the shaft is spinning inside the cylinder coaxially. Basically, torque will be applied to the shaft...

Let $x_{0}=1$ and $x_{n+1}=(-1)^{n}(\frac{\pi }{2}-\arctan(\frac{1}{x_{n}}))$ I have the following options to choose from: 1. $x_n$ is unbounded 2. $x_n$ is increasing and the limit of $x_n$ is $1$ 3. the limit of $x_n$ is $\pi/2$. 4. the limit of $x_n$ is $0$ My attempt: I used...

Homework Statement If possible, calculate the following limit: \lim_{(x,y)\rightarrow (0,0)} {\frac{2x^2 + 3y^2}{5xy}} Homework Equations N/A The Attempt at a Solution [/B] I tried using both parametric and polar equations to find the limit, but neither worked. Setting either x or y...

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x\rightarrow 0} (x-tanx)/x^3$$ I solve it like this, $$\lim_{x\rightarrow 0}1/x^2 - tanx/x^3=\lim_{x\rightarrow 0}1/x^2 - tanx/x*1/x^2$$ Now using the property $$\lim_{x\rightarrow 0}tanx/x=1$$,we have ...

Why the following limit doesn't exists ? $$\lim_{x\rightarrow 0}xe^{-\frac{1}{x}}$$ I think it's because of $\frac{1}{x}$ which doesn't exists, right ?

In Matthew Schwartz's QFT text, he derives the Schrodinger Equation in the low-energy limit. I got lost on one of the steps. First he mentions that $$ \Psi (x) = <x| \Psi>,\tag{2.83}$$ which satisfies $$i\partial _t\Psi(x)=i\partial_t< 0|\phi...

I'm trying to show... $\lim_{{x}\to{0^+}}\left(\frac{e^(\frac{-1}{x})}{x^n}\right)=0$ I guess my calculus is a bit rusty. Can someone help me out? Here's what I've got...

I have the following sequence $(x_{n})$ , $x_{n}=1+\frac{1}{2^{2}}+...+\frac{1}{n^{2}}$ which has the limit $\frac{\pi ^{2}}{6}$.I need to calculate the limit of the sequence $(y_{n})$, $y_{n}=1+\frac{1}{3^{2}}+...+\frac{1}{(2n-1)^{2}}$ I don't know how to start.I think I need to solve the limit...

I came across a question on PSE. I am not sure its a violation to ask the same question here, but there's no answer to the question in there so I wanted to ask it here. Quoting his question,"Since the universe has a positive cosmological constant, there is an upper limit on the mass of the...

Hi! I have the following sequence $$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$ Also I know that $a,b,c\in \mathbb{R}$ and $|c|<1,\ b\neq 1, \ |bc|<1$ I need to find the limit of $x_{n}$. My attempt is in the picture.The result should be $\frac{ac}{(1-bc)(1-c)}$ I...

Hi, $$\lim_{n \to \infty}\dfrac{\dfrac{\ln2}{2}+\dfrac{\ln3}{3}+\cdots+\dfrac{\ln n}{n}}{\ln^2 n}.$$ After I applied Stoltz-Cesaro I got $$\lim_{n \to \infty}\dfrac{\dfrac{\ln2}{2}+\dfrac{\ln3}{3}+\cdots+\dfrac{\ln n}{n}}{\ln^2 n}=\lim_{n \to \infty}\dfrac{\dfrac{\ln (n+1)}{n+1}}{\ln^2...

Hi! $$(x_{n})_{n\geq 2}\ \ x_{n}=\sqrt[n]{1+\sum_{k=2}^{n}(k-1)(k-1)!}$$ $$\lim_{n\rightarrow \infty }\frac{x_{n}}{n}=?$$ I know how to solve the limit but I don't know how to solve the sum $\sum_{k=2}^{n}(k-1)(k-1)!$ which should be $(n! - 1)$ The limit would become $\lim_{n\rightarrow \infty...

<Moderator's note: Moved from a technical forum and thus no template.> How to find this limit? \lim_{x \to 0} \frac{5x} {3 -\sqrt{9-x}} I'd tried to find this limit as below but the result is 0: \lim_{x \to 0} \frac{5x} {3 -\sqrt{9-x}} \lim_{x \to 0} \frac{5x} {3 - \sqrt{9-x}} × \frac{3 +...

Homework Statement lim (1/x - 1/3) / (x-3) x->3 Homework EquationsThe Attempt at a Solution I tried to cancel the bottom (x-3) out by multiplying the top by 3/3 and x/x and then got ((3-x)/3x)/(x-3) but ended with 0/0 and the right answer is -1/9. The top part is confusing me.

I have the following sequence $(a_{n})$, $a_{1}=1$ $$a_{n+1}=\begin{cases} a_{n}+\frac{1}{2} & \text{ if } n \ is \ even \\ \frac{a_{n}}{3} & \text{ if } n \ is \ odd \end{cases}$$ I need to find $$\lim_{n\rightarrow \infty }a_{2n+1}$$ I tried something but I didn't get too far.I rewrite the...

The other day in a fit of boredom I decided to dust off my old math books (high school and undergrad) and see if I can still do basic calculus. These days if I need to solve anything I ask a computer to do it, the hazards of getting a job in industry I suppose. All that said, I have been...

Why $$\lim_{n\rightarrow \infty }\frac{n^{2016}\cdot 2^{n-1}}{3^{n}}=0$$ ? Because $3^{n}> 2^{n-1} $ ?

Dear all, I am trying to solve the following limit: \[\lim_{x\rightarrow 0}(e^{ax}+x)^{\frac{1}{x}}\] where \[a\] is a constant. I know that the limit is equal to \[e^{a+1}\] but not sure how to prove it. Thank you.

I have the sequence from the picture and I have to demonstrate that this sequence has a limit. I always get stuck at this kind of exercises.How to approach an exercise like this?

<Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x \to 0} \cos(\pi/2\cos(x))/x^2$$ I tried to evaluate the limit this way, $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2$$ since $$\cos0=1$$ $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2=\lim_{x \to 0} 0/x^2$$ Now apply...

Homework Statement ##\frac {lim} {x→0} \frac {\sqrt {x+1}-1} {x}## Homework EquationsThe Attempt at a Solution I know the limit as x approaches 0 isn't supposed to be a fraction but I can't get the x approaches 0 under the lim. I couldn't get some of this typed out in latex, it just wouldn't...

I tried to use integration by parts. I took f(x)=arctan(x) => f'(x)= 1/x^2+1 g'(x)=cos(nx) => g(x)= sin(nx)/n So I get sin(nx)/n * arctan(x) - integral from 0 to 1 from sin(nx)/n(x^2+1) How to continue ? I'm always getting stuck with this kind of exercises ( limits of integrals ) because I don't...

$$\lim_{{T}\to{\infty}}N \bar{h}\omega \left( \frac{1}{2} + \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)$$ In term $$\lim_{{T}\to{\infty}}N \bar{h}\omega \left( \frac{1}{e^{\frac{ \bar{h}\omega}{k_BT}}-1} \right)=N \bar{h}\omega \lim_{{T}\to{\infty}}\left( \frac{1}{e^{\frac{...

I have the following equation: x^2 - 2(m+1)x + 3m + 1=0 Also, I know that x1 is the lowest root of this equation. I need to solve lim (x1) as m->infinity A. 1 B. 3/2 C. 0 D. -1/2 E. -1 I tried to solve the equation with the discriminant then to calculate the limit but didn't work. Also, I think...

Homework Statement Hi everyone, I'm currently making my way through Spivak's calculus and got stuck in question 41 of chapter 5. It's important to note that at this point, the book has only reached the subject of limits (haven't reached continuous functions, derivatives, integrals, series...

I've understood the formal definition of limits and its various applications. However, I'm trying to dive more into the history of how the concept of limits were conceived (more than what Wikipedia tends to cover), and how to formally understand and visualise infinitesimals. For example, I know...

Hi, I read the Feynman Lectures Volume 1, Chapter 27, section 27-7, which can be here. In the lecture he describes the fundamental limits of resolution and provides a criterion. Here is the diagram I am referring to, figure 27.-9: There are two light sources, ##P## and ##P'## There is an...

Homework Statement ## \lim x-a \frac {{a^x-{x^a}}}{{x^x}-{a^a}} = -1## then a is? Homework Equations L'Hospital rule The Attempt at a Solution Using LHR we can write numerator as ##\frac{a^x ln{a}-ax^{a-1}}{x^xln(x+1)}## plugging x=a and equating to -1 gives 1-ln(a)=ln(a+1); so 1=ln(a(a+1))...

Homework Statement ## \lim x-0 \frac {xcosx-log(1+x)}{x^2}## Homework Equations ##\frac{log(1+x)}{x}=1## ...(i) The Attempt at a Solution Using (i) we can write numerator as xcosx-x, cancelling x from denominator we have cosx-1/x, this is 0/0 form so we can use LHR which gives us -sinx/1 but...

Homework Statement lim x--->0 |x|^sinx is? Homework Equations lim x-->0 f(x)^g(x), if both functions tend to 0, limit is equal to e^log[f(x).g(x)] with the same limit..(i) The Attempt at a Solution when x>0, it is x^sinx and x<0 it is -1/x^sinx. putting the first case in (i) we get...

If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral. Thank you.