Limits Definition and 1000 Threads

I am self-learning analysis. My steps are as follows, For any ##ε>0##, there is a ##\delta>0## such that, ##|(x^3+2x^2-4x-8) -0|<ε## when ##0 < |x-2|<\delta## Let ##\delta≤1## then ##1<x<3, x≠2##. ##|(x^3-2x^2-4x-8) -0|=|(x-2)(x+2)^2|=|x-2||(x+2)^2| <\delta |(x+2)^2|<19\delta## Taking...

I'm currently taking Math Methods in Physics, and we're working on infinite series right now. In one of the examples, in order to find the limit, they do what you can see below in "Example 3" As you can see, they turn the original equation into a ln equation, so that they can bring the...

I know how to do this using lopital since its a 0/0 indeterminate form. However I would like to do it without using lopital as well..how should I go about it? For starters I thought of rationalizing the numerator and denominator but we cant necessarily apply the (a+b)(a-b) identity since we dont...

I get that the bottom answer isn't a constant - but does this physically represent anything? When I set the two answers equal to each other, I get x = +- 1/sqrt(2) and I am wondering if this represents anything significant. I don't think (mathematically) there is anything wrong with the bottom...

Here's the problem: ##\int_0^{2\pi} \cos^{-1}(\sin(x)) \mathrm{d}x## If I do the substitution ##u = \sin(x)##, both the limits of integration become 0 and the integral would result in 0. But the graph of the function tells me the area isn't 0. Where am I going wrong?

Hi I have to prove the following three tasks I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$ $$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...

Can't attempt to solve the task. I'd appreciate it a lot if you could help!

##\frac{\sqrt{16x^6}-\sqrt{x^2}}{6x^3 + x^2}## ##\frac{4x^3-\sqrt{x^2}}{6x^3+x^2}## ##\frac{4-\sqrt{x^2}}{6+x^2}## My request is may I confirm that I have this correct up to this point? I do know the final answer, I know the suggested process for calculating the answer, but I want to check...

Epsilontic – Limits and Continuity I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are...

Do Psti and Plti values decrease when the measured Pst and Plt (@LV) is transposed to HV?

I have managed to get the answer given by the textbook I'm referencing: 3π (∛4) (1 + 3∛3) However, this took multiple attempts, as I was initially trying to integrate within domain x = 0 - 2. This is the area for the bit that's above the x-axis (y=0 as specified). But the above answer is...

How do I do this from here without using the derivatives of sin or cos ?

Hi, there. I am reading this thesis. On page 146, it reads that I do not know how to calculate the limits when they are viewed as distributions. I am trying to integrate a test function with the limits. So I try (##Q## is defined as ##Q>0##) $$\lim_ {r\rightarrow \infty} \int_{0}^\infty dQ...

I recently read that the “functional information content of human memory" as 10^9 bits at midlife based on testing of text and image retention” It makes sense that anything biological would have a limit but memory is such a strange thing that I can’t see how it would be like storing bytes on a...

For this, Does someone please know why we are allowed to swap the limit as x approaches zero from the right of y with that of In y? Thank you for any help!

(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##. I also tried to calculate the limit directly (not using geometrical argument at...

For this, Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero? Many thanks!

For this, Does someone please know why we are allowed to take limits of both side [boxed in orange]? Also for the thing boxed in pink, could we not divide by -h if ##h > 0##? Many thanks!

I have the following problem and am almost sure of the answer but can't quite prove it: ##f(y)## is nonnegative, and I know that ##\int_0^{\infty } f(y) \, dy## is finite. I now need to calculate (or simplify) the double integral: $$\int_0^{\infty } \left(\int_x^{\infty } f(y) \, dy\right) \...

Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f. If...

##\lim_{x \rightarrow 3} {\frac { \sqrt {9-x^2} } {\sqrt x+\sqrt{3-x}-\sqrt3 }}## I wanted to calculate left hand limit. I find out that the answer is ##\sqrt 6## using GeoGebra. What I have done: I divided the numerator and denominator by ##\sqrt {3-x}##. ##\lim_{x \rightarrow 3^-} {\frac {...

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I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...

I've been bothered lately with an excessive number of freezes and dropouts when trying to conduct meetings on Zoom. As many as 10 freezes with 5 dropouts in 15 minutes. I installed a tool to monitor my connection from my PC. In 4 hours, it gathered the following stats. General stats...

For this problem, The solution is, However, why have they not included limits of integration? I think this is because all the small charge elements dq across the ring add up to Q. However, how would you solve this problem with limits of integration? Many thanks!

Hi PF Searching on the Internet, I've found this definition: Definition: Euler's Number as a Limit (i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}## and (ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}## Questions: 1-Does it make sense...

Hello, I have to compute a double integral of the form ## \int_{0}^{\infty} \int_{0}^{\infty} f(u,v) du dv##, where ##f(u,v)## is not relevant. The following change of variable is advised as a hint: ## u = zt ## and ## v = z(1-t)##. From there, I can reformulate with respect to ##z## and...

The recent high angular resolution images of M87's inner black hole taken by radio telescopes around the globe all linked together in a computer to simulate a giant interferometer, suggest that to achieve still better angular resolution we would need to supplement the terrestrial receiving...

I have been working with some Hypergeometric functions whose behavior I am not quite familiar with. Suppose the equation I wish to analyze is ##p(x) = (e^{x}-1)^{2i}\left({}_{2}F_{1}(a,b;c;e^{x}) + {}_{2}F_{1}(a+1,b+1;c+1;e^{x})\right)## where ##a,b,c## are all complex valued and we have...

Let's us look at the first implication (I will post the reverse implication once this proof has been verified). We have to prove if there is a subsequence of ##(s_n)## converging to ##t##, then there are infinitely many elements of ##(s_n)## lying within ##\epsilon## of ##t##, for any...

In my book it is written "Ends of dipole possesses partial charges. Partial charges are always less than the unit electronic charge (1.6×10−19 C)". Suppose in a double bond(two electron is shared by each atom) or triple bond(three electrons are shared by each atom), can the electronegative atom...

Let ## v_N = \sup \{ s_n : n \gt N \}##. If ## lim \sup s_n = \lim v_N = L##, then for ## \epsilon /gt 0##, we have ##N## such that $$ m \gt N \implies v_m \lt L + \varepsilon$$ $$ n\gt m \implies s_n \lt L + \epsilon$$ Therefore, ## \lim s_n = L##. I don't very much understand limit superior...

In https://www.physicsforums.com/threads/what-assumptions-underly-the-lorentz-transformation.1015982/post-6657920 a discussion evolved from the basic assumptions of the Lorentz transformations, to a paper M. LeBellac, J. M. Levy-Leblond, Galilean electromagnetism, Nuovo Cim. 14B, 217 (1973)...

On one side, if I have any finite value of s = the side of the original triangle of the Koch snowflake iteration, then the perimeter is infinite, so intuitively On the other hand, if I looked at the end result first and considered how it got there, then intuitively (Obviously at n=infinity and...

What prevents making a particle accelerator better than the LHC but only a few centimeters big? After all, you accelerate objects with very small masses. Are there insuperable physical limits? What are the physical limitations?

David Wallace, The sky is blue, and other reasons quantum mechanics is not underdetermined by evidence, Manuscript (2022). arXiv:2205.00568. From the Abstract: ''I argue that there as yet no empirically successful generalization of'' [Bohmian Mechanics and dynamical-collapse theories like the...

It occurred to me that I should ask this to people who passed the stage in which I’m right now, being unable to find anyone in my milieu (maybe because people around me have expertise in other fields than mathematics) I reckoned to come here. Let’s see this sequence: ## s_n =...

Initially '0' is the upper limit and ##a = \frac{Ze^2}{E}## is the lower limit. With change of variable ##x = \frac{Er}{Ze^2}##, for ##r=0##, ##x=0##, and for ##r=\frac{Ze^2}{E}##, ##x=1##, so 1 should be the lower limit. However, he takes 1 as the upper limit, and without a minus sign. Why is...

For values of ##x## such that ##x>0##, is ##\frac{1}{x}## bound above? My reaction is that if ##x>0## then ##\frac{1}{x}## is defined because ##x\neq 0##. But, it is not bound above because ##x## can be taken arbitrarily close to ##0## and ##\displaystyle{\lim_{x \to 0}} \frac{1}{x} = \infty##

do you think there are biological limits to our capacity to understand physics or mathematics?could it be that in the distant future, no scientific progress could be achieved using the human brain, and we have to depend on superhuman level ai to do the work for us?

Consider a Markov chain with state space {1, 2, 3, 4} and transition matrix P given below: Now, I have already figured out the solutions for parts a,b and c. However, I don't know how to go about solving part d? I mean the question says we can't use higher powers of matrices to justify our...

Refreshing..i will attempt part (a) first ...of course this is easy... $$\displaystyle{\lim_{x \to \infty}}\frac{x^2}{e^x} $$ $$\displaystyle{\lim_{x \to \infty}}\frac{2x}{e^x} $$ $$\displaystyle{\lim_{x \to \infty}}\frac{2}{e^x} =0$$

I did only the the first three prop. And on a means we have, on pose or posons means let there be , propriétés means properties, alors meand then. I apologize i am a french native speaker and i am busy so i cannot rewrite this in entirely english

Lim x->c f(x)=L means that for a given ϵ we can find a δ such that when |x-c|<δ-> |f(x)-L|<ϵ. To satisfy the criterion m<f(x)<M we choose ϵ=min (L-m, M-L) and for that ϵ we determine a δ. m<f(x)<M m-L<f(x)-L<M-L |m-L|<|f(x)-L|<|M-L| |L-m|<|f(x)-L|<|M-L| |L-m|<|L|+|m| |f(x)-L|<|f(x)|+|L|...

Suppose a particle is falling under the pull of gravity, the distance it has fallen is given by s=16t^2.Suppose we wish to find the instantaneous speed at t=1. Find the average speed between t=1 and t=1+h where h is any real number except 0. Distance traveled/Time it takes to travel the distance...

I hope I can make this question clear enough. When we have a function such as f(x) = 1/x and calculate the side limits at x = 0, the right side goes to positive infinity. The left side goes to negative infinity. In calculus we are pluggin in values closer and closer to zero and seeing what the...

c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious? First of all I don't think it is obvious but here is an explanation of why the limits are the same. ##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...

I believe the x-axis is vertical here. The graph is composed of i) an infinite number of intervals that start on the ##y=x## line and finish at some ##x## with a decimal expansion ending in ##7\bar{9}##. E.g., from ##0.67## to ##0.67\bar{9}## which is considered ##0.68##. Other examples of...

I’m complete stuck on this problem. I am not sure how to start to find a counterexample to this statement.

Consider item ##vii##, which specifies the function ##f(x)=\sqrt{|x|}## with ##a=0## Case 1: ##\forall \epsilon: 0<\epsilon<1## $$\implies \epsilon^2<\epsilon<1$$ $$|x|<\epsilon^2\implies \sqrt{|x|}<\epsilon$$ Case 2: ##\forall \epsilon: 1\leq \epsilon < \infty## $$\epsilon\leq\epsilon^2...