Limit Definition and 999 Threads

Homework Statement 2. Relevant equation Below is the definition of the limit superior The Attempt at a Solution I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case. I know...

Homework Statement Evaluate the limit as K goes to infinity of s_1,2 (K) Homework EquationsThe Attempt at a Solution Apparently my value for plus the square root is incorrect, apparently the correct answer is 1. Apparently my value for minus the square root is correct, it's negative...

Homework Statement suppose that 0≤xm+n≤xm+xn for all m,n∈ℕ, prove that the limit of xn/n exists when n tends to infinity. Homework EquationsThe Attempt at a Solution I get that xn is bounded by zero and x1. And I guess that xn is monotonous but i find it hard to prove. Or maybe there is...

Homework Statement Consider the standard square well potential $$V(x) = \begin{cases} -V_0 & |x| \leq a \\ 0 & |x| > a \end{cases} $$ With ##V_0 > 0##, and the wavefunctions for an even state $$\psi(x) = \begin{cases} \frac{1}{\sqrt{a}}cos(kx) & |x| \leq a \\...

Homework Statement Let ##f(x) = 0## if ##x## is rational and ##=1## of ##x## is irrational. Prove that ##\lim_{x\to a} f(x)## does not exist for any ##a##. Homework EquationsThe Attempt at a Solution I need help setting this one up. I was thinking that maybe I can argue by contradiction and...

hello I have an exercise which says: Evaluate the following limit. Enter -I if your answer is −∞, enter I if your answer is ∞, and enter DNE if the limit does not exist. limx→0[(1/(7x)−(1)/((e^(7x))−1)] e power 7x when I follow the graph for 1/7x the limit does not exist (goes to...

Say that we are asked to prove, using the definition of limits, that the sequence ##\frac{4n^2+3}{n^2+n+2}## tends to ##4## as ##n## tends to infinity. The following is a screenshot of the solution I found in a YouTube video: (Note that in the definition above, "g" denotes the limit - in this...

Suppose that ##f : \mathbb{R} \to \mathbb{R}## is differentiable at ##a\in\mathbb{R}##. Is it true that if ##\lim_{x\to a}\frac{f(x)-f(a)}{x-a}>0## and ##x>a## then ##f(x)>f(a)##? I'm trying to find a counterexample to show that its false because I think it is, but I'm having a hard tome doing...

Homework Statement Let Tn(x)=1+2x+3x^2+...+nx^(n-1) Find the value of the limit lim n->infinity Tn(1/8).The Attempt at a Solution How do I solve this? I know how to write the polynomial as a series, but not sure how if this is the best way of finding the limit.

I was thinking of generalizing the limit of $\lim_{n\to \infty} (1+x/n)^n=\exp(x)$. What do we know of $$\lim_{n_1\to \infty , n_2 \to \infty , \ldots , n_k \to \infty } (1+\prod_{i=1}^k x_i/n_i)^{\prod_{i=1}^k n_i}$$?

Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...

<Moderator's note: Moved from a technical forum and thus no template.> 1. Homework Statement Is this proof correct? Let K>0, and choose N such that N >= K2, then for all n in the naturals, and n>=N, sqrt(n)+7>=sqrt(N)>=K Is this proof correct? Please tell me

I need a better understanding of limit swtiches and servo motors than I'm getting from Wikipedia :) Any website/textbook recommendations? (My background is physics.) TIA!

Consider the following limit where L'H Rule was correctly applied twice Determine the functions f'(x), g'(x), f(x), and g(x) needed to result in the limit given. \begin{align*}\displaystyle \lim_{x \to 0}\frac{f(x)}{g(x)} \overset{\text{L'H}}=& \lim_{x \to...

Supposedly, infininity has been purged from mathematics. Both the infinitely small and the infinitely large have been replaced by the idea of a "limit." For example, a series x0+x1+x3+... is not considered to be a literal infinite sum with infinite terms but only the limiting value of an...

Hi All I am trying to understand a stress / strain curve for a ductile material. But I am struggling with understanding the difference between the Elastic Limit and the Yield Point. I define these terms as:- Elastic Limit - Is the point on the stress/strain curve where the material will...

Hello! (Wave) I want to check the convergence of the sequences $\left( \left( 1+\frac{1}{\sqrt{n}}\right)^n\right)$, $\left( \left( 1+\frac{1}{2n}\right)^n\right)$. We know that $e^x=\lim_{n \to +\infty} \left( 1+\frac{x}{n}\right)^n$. We have that $\lim_{n \to +\infty} \left(...

Hello! (Wave) Let $(a_n)$ be a sequence of real numbers such that $a_n \to a$ for some $a \in \mathbb{R}$. I want to show that $\frac{a_1+a_2+\dots+a_n}{n} \to a$. We have the following: Let $\epsilon>0$. Since $a_n \to a$, there is some positive integer $N$ such that if $n \geq N$, then...

Limh→0+ (f(rh,h))/h Is the f(rh,h) part the same as f(r+h)-f(h)? I have never seen this before and googling for a long time didn't help, there are no videos with this notation and it's not in my book so, am I just to assume it is? because it doesn't look like it should be the same. Anyone know...

Homework Statement Identify the following limits. Indicate if they do not exist. Assume ##r\ne 0##. ##\displaystyle {\lim_{n\to\infty}}(-1)^n( r^n-r^{-n})## ##\displaystyle {\liminf_{n\to\infty}}(-1)^n( r^n-r^{-n})## ##\displaystyle {\limsup_{n\to\infty}}(-1)^n( r^n-r^{-n})## Homework...

prove the statement using the $\epsilon,\delta$ definition of a limit. $$\lim_{{x}\to{1}}\frac{2+4x}{3}=2$$ so then $$x_0=1\quad f(x)=\frac{2+4x}{3}\quad L=2$$ now $$0<|x-1|<\delta\quad\text {and}\quad\left|\frac{2+4x}{3}-2\right| <\epsilon$$ then...

Homework Statement For each ##n\in\mathbb{N}##, let the finite sequence ##\{b_{n,m}\}_{m=1}^n\subset(0,\infty)## be given. Assume, for each ##n\in\mathbb{N}##, that ##b_{n,1}+b_{n,2}+\cdots+b_{n,n}=1##. Show that ##\lim_{n\to\infty}( b_{n,1}\cdot a_1+b_{n,2}\cdot a_2+\cdots+b_{n,n}\cdot a_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...

Hello. Today I've thinking about limit velocity and speed of ligth. We know that material particles can't achieve that speed, also when the speed of particles increases your own clock walks slowly. In the particular case of ligth your speed don't move anything. This it a explanation of why...

Homework Statement The density of an object is given by its mass divided by its volume: ##p=\frac{m}{V}## Use a calculator to plot the volume as a function of density (##V=\frac{m}{p}##), assuming a mass of 8kg (m=8). In the follow-up question (part b): Evaluate ##\lim_{p \rightarrow 0}...

Homework Statement Evaluate: $$\lim_{θ \rightarrow 0} {\frac{1-cos θ}{sin θ}}$$ Homework EquationsThe Attempt at a Solution By using trigonometric identities, I get to: $$\lim_{θ \rightarrow 0} {\frac{sin θ}{sin θ}}⋅\lim_{θ \rightarrow 0} {\frac{sin θ}{1+cos θ}}$$ By using the Limit Laws, I...

$\textsf{find the value that $\displaystyle \lim_{x \to 0} g(x)$ must have if the given limit statements hold.}$ $$\displaystyle \lim_{x \to 0} \left(\frac{4-g(x)}{x} \right)=1$$ OK the only answer I saw by observation was 2 but the book says it is 4 not sure how you get it with steps

In my text, it states the Basic Limit Results as follows: For any real number ##a##, and any constant ##c##, (i) ##\lim_{x \rightarrow a}{x}=a## (ii) ##\lim_{x \rightarrow a}{c}=c## Now from the previous chapter, I am used to seeing these as taking the limit of some function as the x values...

Homework Statement Find the following limit: $$\lim_{x \rightarrow 10}\frac{x-10}{4-\sqrt{x+6}}$$ Homework EquationsThe Attempt at a Solution [/B] Please see attached work. I have a few questions (other than if my solution is correct or not). First, is at step (ii)(C): What makes me uneasy...

Homework Statement Find the slope of ##y=x^2+4## at (-2,8) and the equation for this line. Homework EquationsThe Attempt at a Solution This problem is intended to give an intuition on how limits work and I think I get the general idea. If we want to find the rate of change (or slope) of some...

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]

Hi guys. I'm using the Monte Carlo method to simulate a spin lattice. If I have a square lattice, L x L, I can plot the phase transition temperature by the inverse of the lattice length (1/L) to find the phase transition temperature in the thermodynamic limit (extrapolating the curve for 1/L =...

Hi guys. Anyone knows a article showing the method of extrapolation curve of the phase transition's temperature by the inverse of lattice size, applied at low-dimensional lattices, like nanotube and nanowire, for example? Thanks a lot!

Homework Statement Solve the ##\lim_{n \rightarrow +\infty} \sqrt [n] \frac {n²+1} {n⁷-2} ## 3. The attempt of a solution: First I thought about using L'Hopital's rule, but the nth root makes it useless. Then I thought about to eliminate the root multiplying it by something that is one, but...

Do we know enough of the workings of string theory to say what factors give rise to a large or small value of the velocity of propagation of massless fields for a given multiverse? Thanks!

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...

##\displaystyle \lim_{a \to 0^+} a^2 \log a = 0 \cdot (- \infty)##, which is an indeterminate form. So ##\displaystyle \lim_{a \to 0^+} a^2 \log a = \lim_{a \to 0^+} \frac{\log a}{a^{-2}} = \lim_{a \to 0^+} \frac{\frac{1}{a}}{(-2)a^{-3}} = -\frac{1}{2}\lim_{a \to 0^+} a^2 = 0##. Is this correct?

The Klein-Gordon equation has the Schrodinger equation as a nonrelativistic limit, in the following sense: Start with the Klein-Gordon equation (for a complex function ##\phi##) ## \partial_\mu \partial^\mu \phi + m^2 \phi = 0## Now, define a new function ##\psi## via: ##\psi = e^{i m t}...

Is there a limit to how steep a refractive index gradient can be before ray optics are no longer able to predict the path of the light? How is it related to wavelength? Under what conditions the light will be able to travel perpendicular to the gradient In a straight line? (having diffrent index...

Let f:\mathbb{R}^m\rightarrow\mathbb{R}^m. Define the zero set by \mathcal{Z}\triangleq\{x\in\mathbb{R}^m | f(x)=\mathbf{0}\} and an \epsilon-approximation of this set by \mathcal{Z}_\epsilon\triangleq\{x\in\mathbb{R}^m|~||f(x)||\leq\epsilon\} for some \epsilon>0. Clearly \mathcal{Z}\subseteq...

Hey! :o Let $u(x,t), A(x)$ be functions, for which holds the following: We have the pde $u_t+a(u)u_x=0$. Let $A'(u)=a(u)$ then the pde can be written as $u_t+A(u)_x=0$. We have the following integrals $$\int_{a-\epsilon}^au\cdot \left (\frac{x-a}{\epsilon}+1\right )\...

i'm trying to review calculus and look a little deeper into proofs/derivations/etc. I'm doing this both for fun and to review before i go back to school. am i the only one who has difficulty understanding the "rigorous" definition of the limit? i found this web page...

Challenge Problem: Let $A$ be an $r \times r$ matrix with distinct eigenvalues $λ_1, . . . , λ_r$. For $n \ge 0$, let $a(n)$ be the trace of $A^n$. Let $H(n)$ be the $r \times r$ the Hankel matrix with $(i, j)$ entry $a(i + j + n - 2)$. Show that $ \displaystyle \lim_{n \to \infty} \lvert...

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...

Disclaimer: to avoid giving the impression of speculative nature, I state the purpose of this thread is only to conflate known theory with my own understanding in a specific point and clarify where the disagreement lies; that is all. TOV limit: since early research in black hole (BH) formation...

https://photos-5.dropbox.com/t/2/AAC1PAsxThHE7dTxxumANssxIDSrZGA0wi9u1T2alieA9g/12/217355121/png/32x32/1/_/1/2/Screen%20Shot%202018-04-24%20at%2014.40.53.png/EJ6fyaMBGOQEIAIoAg/zVJasOZ8quUZpWc6eN6tzuO7YSmC-VjpQ4ikXIkpC8A?preserve_transparency=1&size=2048x1536&size_mode=3 So in looking at the...

When we define a limit of a function at point c, we talk about an open interval. The question is, can it occur that function has a limit on a certain interval, but it's extension does not? To me it seems obvious that an extension will have the same limit at c, since there is already infinitely...

Homework Statement [/B] We have a function f(x) = |cos(x)|. It's written that it is piecewise continuous in its domain. I see that it's not "smooth" function, but why it is not continuous function - from the definition is should be..Homework Equations [/B] We say that a function f is...