Limit Definition and 999 Threads

Hello,This is actually a piece of a little bigger problem (convergence of a series) - you can see the ratio test ak+1 / ak That's why the (n) and (n+1) terms I have lim n->∞ of (n√n) / (n+1)√(n+1) ∞/∞ I have tried L'Hopitals rule (requiring multiple times) and I am not seeing an end...

Hi, can anybody help me with this two limits? I have to prove them by the definition of limit. Thank you in advance.

How do you solve this kind of limit? \lim_{x \to ∞} \left( \frac{x^2 - 1}{x^2 + 2x + 5} \right)^{-2x} Please give me a clue.

I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ... I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...

(NOTE: I have had a few similar postings lately on this subject, but they were much broader in scope, so I am posting only for this particular case; everything else has been figured out.) If given that limx -> a f( x ) = +∞ limx -> a g( x ) = +∞ what is the epsilon-delta formulation for...

What is exactly the definition of symmetric limit? It's the first place in the book that I see this notation, and they don't even define what it means. How does it a differ from a simple limit or asymptotic limit? I found a few hits in google, but it doesn't seem to help...

I’ve read many Legends and Canon Star Wars books and I always take away stuff on their limits of technology and science. Over the years; here are some things they said science can’t do. 1.) Cybernetic liver- In Lost Stars, it was said Ciena’s liver could not be replaced as it was one of the...

I have the function: ##\sqrt{\left(\frac{x}{h}+1\right)^{2}+\left(\frac{y}{h}\right)^{2}}-\sqrt{\left(\frac{x}{h}\right)^{2}+\left(\frac{y}{h}\right)^{2}}## I would like to find an analytical solution, the equivalent function, in the limit of h approaching zero.Additional info which might be...

A 2019 paper by Gorecki et al. derives an uncertainty principle limit that is larger than the conventional Heisenberg limit by a factor of ##\pi##: https://arxiv.org/abs/1907.05428 I'm wondering if any QM experts have seen this and what your thoughts are.

Hello all, Given following limits: ##\lim_{x \rightarrow 1} {\frac {\sqrt x -1} {x^2 - 1}}## ##\lim_{x \rightarrow 1} {\frac {\sqrt {x+1} - 2} {x - 3}}## ##\lim_{x \rightarrow 1} {\frac {\sqrt[3] x - \sqrt[4] x} {\sqrt[6] x - \sqrt x}}## Those limits can be evaluated by letting ##x = t^2##...

How can i get rid of the last delta x

Someone asked me what a dime was (this is UK.) I, not knowing, nevertheless promptly replied, it must be 5 cents, because they called their 55mph speed limit the double dime. Then of course went to Google to check and found that it is 10 cents. How does double 10 become 55? So back to Google...

Hey! :o I want to check the existence of the limit $\lim_{x\to 0}\frac{x}{x} $ using the definition. For that do we use the epsilon delta definition? If yes, I have done the following: Let $\epsilon>0$. We want to show that there is a $\delta>0$ s.t. if $0<|x-0|<\delta$ then...

When does an on/off ratio become impractical? I've read that typical CMOS devices have an on/off ratio of 10^6-10^10. Is there a lower limit that prevents logic devices from functioning appropriately? As in, what is a practical lower limit for the on/off ratio but still enable logic functionality?

I wanted to know what is understood as the dispersive approximation (or limit) in the context of the Jaynes-Cummings model for one mode of the field.

$$\lim_{x\to\infty} (x^3+x^2 +\frac{x}{2})-x^3\sqrt{(1-\frac{1}{x^6})} = \lim_{x\to\infty} x^3+x^2+ \frac{x}{2} -x^3=\lim_{x\to\infty} x^2 + \frac{x}{2} = \infty. $$ is this it?

Would this ##f'(x)>g'(x)\,\forall x\in [a,\infty)\text{ and }f,\,g\underset{\infty}{\to}0\Rightarrow \lim_{x\to\infty}f(x)/g(x)=0## hold if ##\frac{f(x)}{g(x)}\neq c##?

Hello all, I am trying to solve a limit: \[\lim_{x\rightarrow 0}\frac{sinh (x)}{x}\] I found many suggestions online, from complex numbers to Taylor approximations. Finally I found a reasonable solution, but one move there doesn't make sense to me. I am attaching a picture: I have marked...

Hello everyone, I want to calculate the following limits: \[\lim_{x\rightarrow \infty }\frac{[x\cdot a]}{x}\] using the sandwich rule, where [xa] is the integer part function defined here: Integer Part -- from Wolfram MathWorld I am not sure how to approach this. Any assistance will be...

I'm not sure if the way I solve these limits is correct, so let me know if I'm doing something wrong. $$\lim_{x^2+y^2 \rightarrow +\infty} {\frac {xy} {x^2+y^2}}$$ $$r = x^2+y^2$$ $$\lim_{r \rightarrow +\infty} {\frac {r\cdot cos(v) \cdot r \cdot sin(v)} r}$$ $$\lim_{r \rightarrow +\infty}...

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}e^{-n}\sum^{n}_{k=0}\frac{n^k}{k!}$

I am currently starting with my first qft lectures and i am trying to see for the free particle that the propagator $$ <x_i | e^{-i\frac{p}{2m} T|x_f}>$$ will equal to one if x_f = 1, x_i=0 m=1 u=1 p=1, T=1 and $$\hbar \rightarrow 0$$ or 0 otherwise. I understand that this limit will result in...

Evaluation of $$\displaystyle \lim_{n\rightarrow \infty}\sum^{n}_{k=1}\bigg(\frac{k}{n^2}\bigg)^{\frac{k}{n^2}+1}$$

As I have studied before, I found that Infinite Red Shift occurs where gtt = 0 but this exercise says that on Kerr's Black Hole it doesn't really work like that. Right now I'm blocked because I didn't find anything on the internet about it so I don't know how to show this phenomenon. Any help...

In thermodynamics limit, does function of many mean values(of some physical quantities) equal mean value of the function of the values?

I was reading that in inverse scattering approach, we divide the region of interest into discrete grids and size of each grid should be much smaller than the incident wavelength (usually smaller than one-tenth of wavelength). By this logic, theoretically, I can use inverse electromagnetic...

but plugging -2 you clearly get a 0/0 answer. which one is correct? is x=-2 an asymptote?

I don't know what do do from here other than i can make the 3/e^x a 0 due to the fact its divided by such a large number. What do i do with the e^-3x? Thanks for the help

We have so many great books available for Calculus, such as : Spivak's Calculus, Stewart Calculus, Thomas Calculus , Gilbert Strang's Calculus, Apostol's Calculus etc. These books are very nice but they teach you the concepts well and all the standard techniques that are available for solving...

The Propane industry mandates that a tank not be filled more than 80%. The question I have is this: how do I calculate the limit of liquid propane in a standard 3800 liter tank given a 30 degree rise in temperature (from 273 K to 303 K) such that it will not rupture the tank? For example, can I...

What does the reconstructed wave look like if we sample the input an infinitesimal amount under the Nyquist limit? I can intuitively picture how we can (ideally) reconstruct an input sampled at the Nyquist limit (and appropriate phase) because we are able able to get the extreme values of the...

*Given: δ = |all real numbers| *Given: ϵ = |all real numbers| For any x value within +/- |δ| of c, we can find a y=f(x) within the corresponding +/-|ϵ| of L. According to my professor, the mathematical representation of this is |x-c| < δ and |f(x) - L | < ϵI fail to understand why it cannot...

Find the limit as x approaches 0 of x2/(sin2x(9x)) I thought I could break it up into: limit as x approaches 0 ((x)(x))/((sinx)(sinx)(9x)). So that I could get: limx→0x/sinx ⋅ limx→0x/sinx ⋅ limx→01/9x. I would then get 1 ⋅ 1 ⋅ 1/0. Meaning it would not exist. However the solution is 1/81...

Are electrons limited to how much energy they can carry(if that term can be used)?

Show the following limit will converge to delta(y-b), lim 1/|a| f_x((y-b)/a)=delta(y-b) a-->0

I'm getting 2119.36 for the concentration of mg/cubic meter of this substance...it just feels wrong though. Steps I followed: First, I figured out how many grams of the substance there were using the density formula, then saw how many were present per cubic meter after calculating the volume...

If one approaches the origin from where ##x_2=0##, the terms ##x^2_1x_2+x^2_2x_3## in the denominator equal ##0##. Substituting ##|\textbf{x}|^2## for ##t## yields the expression ##\frac{e^t-1}{t}##, which has limit 1 as ##\textbf{x}\to\textbf{0}## and thus ##t\to0##. So the limit should be 1 if...

Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...

Consider limx→3x^2=9. Find a maximum value of δ such that: |x2 - 9|<0.009 if |x-3|<δ I just learned how to do this today and I am quite comfortable doing this if the function is linear, however now I am struggling with working with quadratic functions. So far this is what I have come up with...

I don't know how to show that this limit is zero. It seems that ##\sum_{i=1}^N a_{i,N} /N = 1## and the fact that ## 0 < a_{i,N} < M > 1## implies that some ##a_{i,N}## are less than one. Another conclusion I guess is correct to draw is that ##\lim_{N \to \infty} \sum_{i=1}^N a_{i,N}^2 /N < 1##.

Instinct tells me to just plug in the number, say the limit is zero, and be done with it. But at the same time, while reading the statement from the "Relevant equations" section of this post, I cannot feel but feel some doubt as to whether or not this is the right approach. I mean, only the...

Let's say you are rubbing a balloon on your hair to make it charged. If you then discharge the balloon and rub it on your hair again (and repeat this process numerous times). Would your hair run out of electrons so eventually you would be unable to charge the balloon, or would your hair gain...

Find \[\lim_{n\rightarrow \infty}\int_{0}^{\infty}\frac{e^{-x}\cos x}{\frac{1}{n}+nx^2}dx.\]

This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary. PRELIMINARY: Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...

The classical limit of QM that have always puzzled me. There are common statement saying that you can recover classical mechanics by taking the limit of h->0 or by taking large quantum numbers. Other times times the Erhenfest theorem or the Madelung/hydrodynamics version of the Schroringer...

Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...

lim 2x + 3 = ∞. x→∞ Pretty intuitive when considering the graph of the function. But how would I show this using the epsilon-delta definition?Thanks!

I simplified somewhat and got (1/e-(1-x/(1+x))x)/(1/x) So i can't find that it is 0/0 form so tried by applyying L'Hospitale,But it just became complicated.So need help.

I want to show that the limit of the following exists or does not exist: When going along the path x=0 the limit will tend to 0 thus if the limit exists it will be approaching the value 0 when going along the path y=0, we get an equation with divisibility by zero. Since this is not possible...

Let's say I have a function whose derivative is (tan(x)-sin(x))/x. It is not defined for X=0 but as X approaches 0 the derivative approaches 0, so should I conclude that my function is not differentiable at X=0 or should I conclude that the derivative at X=0 is 0.