Limits Definition and 1000 Threads

Homework Statement Please see attached image for the full scope of the problem, and to see the work drawn out by the text. My question lies with line 3 as it is clear that u-substitution was used on a definite integral, but the limits of integration were not changed. Homework EquationsThe...

This is a random problem I am trying to figure out. The context doesn't matter. I wish to define a function z(x, y) based on the following limits: 1. lim z (x→∞) = 0 2. lim z (x→0) = y 3. lim z (y→∞) = ∞ 4. lim z (y→0) = 0

Hi, I got curious on the limits of variatonal principle. As far as i know all of the theories can be reformulated as problem of finding extremum of some action. Not only that, but it seems to be most convenient method for looking for new theories in a lot of cases. So my question is, what are...

My code version is 2.7 I have a disk source of R=0.3 cm, 60 cm above in z axis. I want set limits for the x and y axis, but, I can only put one command "axs" and "ext". How can i define two limits with one command? my code it is like this SDEF pos=0 0 60 rad=d1 axs=1 0 0 ext=d2 PAR=2 ERG=0.018...

A pre-print of a conference paper from eleven months ago analyzes the extent to which the available data on the CKM matrix element values rules out beyond the Standard Model Physics. It finds that in the most rigid model dependent analysis, that new physics are excluded up to a characteristic...

Hi. I came across the following integral in contour integration lim(ε→0) "integral of" exp(iaεeiθ) dθ = θ If I take the limit first then it just becomes the integral of 1 which is θ. I have 2 questions - If I take the limit first and then perform the integral do I always get the same answer as...

Homework Statement Evaluate: ##\lim_{x \rightarrow -\infty} {\frac{3x^3+2}{\sqrt{x^4-2}}}## Homework EquationsThe Attempt at a Solution For limits involving fractions, it's a good idea to divide the numerator and the denominator by the highest degree x in the fraction. In doing this, we can...

Homework Statement Let ##(a_n)## be a arbitrary real sequence. Given that the sequence ##\frac{a_{n+1}}{a_n}## is convergent, show that ##\lim \frac{a_{n+1}}{a_n} = \lim \frac{a_n}{a_{n-1}}## Homework Equations Take ##\mathbb{N} = \{1,2,3, \dots\}## The Attempt at a Solution In general, I...

The theorem that allows one to distribute the limit over addition is the following: Let ##(a_n), (b_n)## be sequences that converge to ##L## and ##M## respectively. Then ##\lim (a_n+b_n) = L + M##. So evidently, a hypothesis of distributing the limit is that we know ##a_n## and ##b_n##...

Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions. I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture. Now my question: We are taking the limit of a composition of...

Homework Statement Find y; $$y=\lim_{x \rightarrow 0} {\frac {1} {x^2}-\frac{1}{tan^2(x)}}$$ Homework Equations $$\lim_{x \rightarrow 0} {\frac{tan(x)}x}=1$$ $$\lim_{x \rightarrow 0} {\frac{sin(x)}x}=1$$ The Attempt at a Solution \begin{align} y & = \lim_{x \rightarrow 0} {\frac {1}...

Homework Statement lim x~∞ 〈√(x⁴+ax³+3x²+ bx+ 2) - √(x⁴+ 2x³- cx²+ 3x- d) 〉=4 then find a, b, c and d[/B]Homework Equations all the methods to find limits The Attempt at a Solution it can be said that the limit is of the form ∞-∞.I am completely stuck at this question.the answer is a=2...

Homework Statement lim x~a 〈√(a⁺2x) -√(3x)〉 ÷ 〈√(3a+x) - 2√x〉[/B]Homework Equations rationalisation and factorisation[/B]The Attempt at a Solution i had done rationalisation but the form is not simplifying.pleasez help me.[/B]

Homework Statement if f(x)= lim(n→∞) e^(xtan(1/n)log(1/n)) and ∫f(x)/(sin^11x.cosx)^1/3 dx=g(x)+c, then 1) g(pi/4)=3/2 2) g(x) is continuous for all x 3) g(pi/4)= -15/8 4) g(pi/4)=12 2. The attempt at a solution Part a-Evaluating the limit, since 1/n tends to 0, log(1/n)→-∞=-n, using...

Homework Statement https://gyazo.com/268bef206850bfbf30fb0cca3f783599 <----- The question Homework EquationsThe Attempt at a Solution Had this on a test today, honestly not sure how to evaluate. I know you can pass the limit to the inside of arctan but I can't see how the inside goes to...

I have the integral ##\displaystyle \int_0^{2 \pi} \frac{1-\cos x}{3+\cos x} ~ dx##. I want to make the tangent half-angle substitution ##t = \tan (x/2)## so that I can get a rational function. However, both limits of integration just become zero. This is the first case. In the second case, I...

Interesting economic paper on the limits of bitcoin: The amount of computational power devoted to anonymous, decentralized blockchains such as Bitcoin’s must simultaneously satisfy two conditions in equilibrium: (1) a zero-profit condition among miners, who engage in a rent-seeking competition...

So we know that we typically have to use epsilon delta proofs for determining a limit of a multivariable function because there are infinite paths. But can we use removable discontinuities to prove a limit? Say we want to evaluate the lim( x^2-y^2)/(x+y) as (x,y)->(0,0). we can factor as...

The basic concept is to have your space probe(s) - likely nanocraft [1] on a spinning object in space which allows you to preserve the momentum you give it while accelerating it faster. Then once you are at a speed you can simply release the nanocraft in the direction you want it to go in. More...

Debye assumed sound wave dispersion relation for phonons(##ω=vK##) and this corresponds to acoustic modes in low frequency limits. That's why it explains low temperature heat capacity fairly well. But how could this also explain high temperature limit(##C=3k_B## per atom)? I know Debye...

Homework Statement Please look at the photo! Homework Equations -x^2+4x=x^2-6x+5 The Attempt at a Solution I got 2x^2-10x+5 but it says it's wrong

\begin{align}\displaystyle &=\int_{0}^{8}\displaystyle \int_{\sqrt[3]{x}}^{2} \frac{dydx}{y^4+1}&&(1)\\ &\qquad D: 0\le x \le 8, \quad \sqrt[3]{x}\le y\le 2 &&(2)\\ &=\int_{0}^{2}\int_{0}^{y^3} \frac{1}{y^4+1} \, dxdy&&(3)\\ &=\int_{0}^{2}\frac{y^3}{y^4+1} \...

Homework Statement a. Compute the limit for f(x) as b goes to 0 Homework Equations $$f(x) = \frac{(a+bx)^{1-1/b}}{b-1}$$ ##a \in R##, ##b\in R##, ##x\in R## The Attempt at a Solution ##a+bx## goes to ##a## ##1/b## goes to ##\infty## so ##1-1/b## goes to ##-\infty## ##(a+bx)^{1-1/b}## then goes...

Some time ago I was playing with the oscillator when I noticed a few funny things. Consider first the 1D oscillator with Hamiltonian $$ \displaystyle H(q,p) = \frac{p^2}{2m} + \frac{m\omega^2}{2}q^2$$ whose solutions are $$ q(t) = q_0cos(\omega t) + \frac{p_0}{m\omega}sin(\omega t), p(t) = m...

Homework Statement Let ##x\in\Bbb{R}## such that ##x\neq 0##. Then ##x=LIM_{n\rightarrow\infty}a_n## for some Cauchy sequence ##(a_n)_{n=1}^{\infty}## which is bounded away from zero. 2. Relevant definitions and propositions: 3. The attempt at a proof: Proof:(by construction) Let...

Could someone check if my answers are right and help me with question 5?

Suppose I have the sequence ##a_n = 2^{(-1)^n}##. So ##\displaystyle (a_n) = (\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,\frac{1}{2},2,...)##. Clearly, this sequence has two subsequential limits, ##\displaystyle \{\frac{1}{2},2 \}##. This clear from observation, but I'm not sure how I can be sure...

I have a question, not based on any homework but just based on my own readings. If ##L \in \mathbb{R}## and ##L>0##, and if ##\lim a_n < L##, does there necessarily exist an ##N \in \mathbb{N}## such that ##a_N < L##? How would I prove this if its true? I tried to use the definition of...

If I have monotonic sequence, would it suffice to analyze |a(n)-a(n-1)| as n gets large? I know for Cauchy sequences, you have to analyze every term after N, but for monotonic sequences that are also Cauchy, can you just analyze the difference between consecutive terms?

Homework Statement "Let ##E \subset ℝ##. Prove that ##E## is closed if for each ##x_0##, there exists a sequence of ##x_n \in E## that converges to ##x_0##, it is true that ##x_0\in E##. In other words, prove that ##E## is closed if it contains every limit of sequences for each of its...

Suppose that $\int_{-\infty}^{\infty} f(x)\,dx$ converges. Then $\lim_{{x}\to{-\infty}}f(x) = \lim_{{x}\to{\infty}}f(x)$. Why is it true? I have some trouble understanding this intuitively.

I would like some help to find some additional info on generalized functions, generalized limits. My aim is to understand the strict definition of delta dirac δ(τ).If you could provide a concise tutorial focusing on δ(τ) not the entire theory...it would be of great help. I am not a math...

Homework Statement I want to change the integration limits of an integral in cylindrical to cartesian coordinates. For example the integral of function f(r) evaluated between b and R: ∫ f(r)dr for r=b and r=R (there is no angular dependence). For write de function in cartesian coordinates...

In the book " Real Analysis: Foundations and Functions of One Variable" by Miklos Laczkovich and Vera T. Sos, Theorem 5.2 (Chapter 5: Infinite Sequences II) reads as follows:https://www.physicsforums.com/attachments/7722 Can someone inform me if there is an equivalent theorem that holds in...

Homework Statement [/B] The proposition that I intend to prove is the following. (From Terence Tao "Analysis I" 3rd ed., Proposition 6.1.7, p. 128). ##Proposition##. Let ##(a_n)^\infty_{n=m}## be a real sequence starting at some integer index m, and let ##l\neq l'## be two distinct real...

If one considers the quantized levels of E, for the solutions to the Schrödinger eqn,, then I am wondering: what are the lowest possible energies that can occur for the Schrödinger eqn? I take the highest possible energy is at the classical limit, but is the zero-point energy the absolute...

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of the proof of Lemma 1.3.3 ... Duistermaat and Kolk"s proof of Lemma 1.3.3. reads as follows:In the above proof...

Hi PF! Suppose I have two functions ##f(x),\,g(y)## that are numerically defined as vectors (i.e. ##g(y) = [0,1,4,9,16]:y = [0,1,2,3,4]## and say ##f(x) = [0,1,8,27,64]:x = [0,1,2,3,4]##) and am trying to compute $$\int_0^1 f(x) \int_x^1 g(y)\, dydx.$$ How would I do this in MATLAB? I could be...

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...

We know that belt drives are limited in their max possible power rating and most high power/torque applications(trucks) use gear drives. I wanted to know the main factor that limits the power/torque rating of belts. Is it Frictional slip or Belt material? If the frictional slip could be...

Particle Data Group - 2017 Review has some strong lower limits for the mass scales of possible quark and lepton compositeness, or at least the compositeness of the easier-to-study ones, like up and down quarks and also electrons. The limits are well into the TeV range, though they are somewhat...

Hey! :o I want to show that $\displaystyle{\lim_{x\rightarrow \infty}\frac{e^x}{x^{\alpha}}=\infty}$ and $\displaystyle{\lim_{x\rightarrow \infty}x^{\alpha}e^{-x}=0}$ using the exponential series (for a fixed $\alpha\in \mathbb{R}$). I have done the following: $$\lim_{x\rightarrow...

Homework Statement [/B] Trying to understand this limit: where ##r>0## Homework Equations [/B] I think it's best to proceed by writing this as: ## N=1 \pm \frac{\sqrt{Ae^{2rt}}}{\sqrt{1-Ae^{2rt}}} ## The Attempt at a Solution [/B] since ##r>0 ## the exponential term ##\to ## ##\infty##...

Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I can't find notes on it.It think it has something to do with limits. Many thanks.

Hi, I was looking for a symbol in math that is commonly applied when a limit to a function does not exist. Is there such a symbol? I could not find any.

Hello . I have problems with two exercises . 1.\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} } Here, I have to write when this limit will be exist. 2.\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} } Here, I have to write for which a \in \mathbb{R}_+ this...

So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...

Homework Statement Integrate ∫ (tan √x) / (2 √x) dx Homework Equations Limits from 0 to ∞ The Attempt at a Solution Put u = √x du/dx = 1/ (2 √x) dx = du * (2 √x) now question becomes ∫ tan u du = log sec u = log (sec √x) now applying limits ∫ tan u du = log (sec √∞) - log (sec √0) = log...

Given that 0 < sin x < x is true for 0 < x < π/2. From the above, can we conclude that 0 < sin (x/2) < x/2? How about 0 < sin (x/5) < x/5? Why? How about 0<sin 3x < 3x ? Why?

Homework Statement Evaluate the following limit or explain why it does not exist: limx→∞ 24x+1 + 52x+1 / 25x + (1/8)6xThe Attempt at a Solution I know there is the method where you divide through by the highest term in the denominator, but can that be applied here?