Epsilon Definition and 215 Threads

Can someone please explain Epsilon delta definition of limit in detail?

Hi there, I'm having trouble understanding how to prove things using the \delta \epsilon definition. I have read a few other threads and sites, but I can't seem to put it together. For example, I came across this problem, if given limx-->af(x) = L, how would I prove (using delta-epsilon and...

Homework Statement For the limit below, find values of δ that correspond to the ε values. Homework Equations epsilon = .5 and epsilon = .05 The Attempt at a Solution These kinds of problems do you have to use a graphing calculator to figure it out? for epsilon = .05 |(9x + x...

Homework Statement evaluate lim2x^2 as x approaches 3 using formal definition (epsilon and delta) of limit Homework Equations The Attempt at a Solution

Im having trouble proving limits of multivariable functions. I understand the principal behind delta-epsilon proofs but I can't get it to work. Once I set up the inequalities I am stuck. The only example in my book seems very convenient though . 3x^2 * abs(y) divided by x^2 + y^2...

i know how to do basic proofs, but some proofs on the actual limit theorems confuse me. my textbook's choices for delta are very obscure and i have no idea how they even came up with them. for the proof of the limit theorem where the limit of a product of 2 functions is equal to the product...

Homework Statement Let K be a compact sebset of a metric space (X, d) and let \epsilon greater than 0. Prove that there exists finitely many points x_1 x_2, ... x_n \in K such that K is a subset of the union of the \epsilon neighborhoods about x_i Homework Equations N/A The Attempt...

hey if lim (x-->0) f(x) = L where 0 < |x| < d1 implies |f(x) - L | < e how do i prove lim (x --> 0) f(ax) = L? i know 0 < |ax| < |a|d1 d2 = |a|d1 but the textbook says d2 = d1/|a| help you guyssssssssssssssssssssssssssssssss

Given: limit of (sin x)/x as x --> 0 and that ε = .01 Problem: Find the greatest c such that δ between zero and c is good. Give an approximation to three decimal places. Equations: 0 < |x - a| < δ 0 < |f(x) - L| < εAttempt: 0 < |x - 0| < δ 0 < | sin(x)/x - 1| < ε 0 < | sin(x)/x - 1| < .01 0...

Given the limit of \frac{x^2+2x}{x^2-3x} as x approaches 0 equals \frac{-2}{3} and that ε = .01, find the greatest c such that every δ between zero and c is good. Give an exact answer. 0 < |x-0| < δ 0 < |\frac{x^2+2x}{|x^2-3x} + \frac{2}{3|}| < ε |\frac{x(x+2)}{|x(x-3)} +...

Homework Statement Let f: \Re \rightarrow \Re and g: \Re \rightarrow \Re be functions such that lim_{x \rightarrow 1} f(x)=\alpha and lim_{x \rightarrow 1} g(x)=\beta for some \alpha, \beta \in \Re with \alpha < \beta . Use the \epsilon-\delta definition of a limit to prove...

Homework Statement given a function defined by f(x,y) {= |xy|^a /(x^2+y^2-xy), if (x,y) cannot be (0,0) and = 0, if (x,y) = (0,0) Find all values of the real number a such that f is continuous everywhere e= epsilon d= delta In order to prove this, I know I need to do an...

Okay so here's the limit lim(as x --> -1) 2x^2=2 I have |2x^2-2| < 1 2|x + 1||x - 1| < 1 |x + 1||x - 1| < 1/2 and 0 < |x + 1 | < delta -1 - delta < x < -1 + delta |x - 1| < -2 + delta |x + 1| < 1/2(-2 + delta) so is delta = 1/2(-2 + delta) correct...

Homework Statement Prove the follow statements directly using the formal \epsilon , \delta definition. \lim_{x\rightarrow 1} \frac{x + 3}{x^2 + x + 4} = \frac{2}{3} Homework Equations The Attempt at a Solution 0 < |x - 1| < \delta \rightarrow 0 < |\frac{x + 3}{x^2...

Homework Statement Suppose A_n is, for all natural numbers n, some finite set of numbers in [0,1] and A_n intersect A_m={ } if m!=n Define f as follows: f(x) = 1/n if x is in A_n and 0 if x is not in A_n for all n. Prove that the limit as x goes to a of f(x) = 0 for all a in [0,1]...

Hi, Want to show: a<=r However I ended up with such an argument: For each eps>0, a<r+eps. Does this statement imply a<=r?

A positive number epsilon (e) and a limit L of a function f at a are given. Find delta such that |f(x)-L|< epsilon if 0 < |x-a| < delta. \lim_{x->5}, 1/x= 1/5, \epsilon=.05. That implies the following |\frac{1}{x}-\frac{1}{5}|< \epsilon \mbox{ if }|x-5|<\delta. Which implies...

I have started studying maths on my own using a University maths book that may not lend itself to self study. So I was hoping someone could help me with the following.abs{sqrt{x}-2} < .05 if 0 < abs{x-4} < delta. I rewrite this as abs{sqrt{x}-2} < .05 if abs{(sqrt{x}+2)(sqrt{x}-2)} < delta...

Here's an example that helps illustrate my question: Prove: A sequence in R can have at most one limit. Proof: Assume a sequence {xn}n\inN has two limits a and b. By definition: -For any \epsilon>0, there exists an N\inN such that n\geqN implies that |xn-a| < \epsilon/2. -A...

I have an example bit I can't quite follow it...? Use epsilon -delta definition of continuity to prove f(x) = 3x^2 - x is continuous at x=2 Ep > 0 and delta > 0 in terms of Ep f(x) -f(2) = 3x^2 - x -(3*2^2 -2) f(x) - f(2) = 3x^2 -x - 10 f(x) - f(2) = (3x + 5)(x - 2) So far so...

Like many people on this forum, i am seemingly having a lot of trouble grasping the concepts of Epsilon Delta proofs and the logic behind them. I have read the definition and i realize for e>0 there is a d>0 such that... 0<sqrt((x-1)^2 - (y-b)^2) < d then f(x,y) - L <e (excuse my use of...

I understand most of the logic behind the formal definition of a limit, but I don't understand the the logic behind an epsilon delta proof. The parts I'm having trouble with are these: 1. How does proving that, the distance between the function and the limit is less than epsilon whenever the...

The epsilon delta rule states \epsilon_{ijk}\epsilon_{pqk}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp} I am constantly using this but get stuck when it is applied. For example \epsilon_{ijk}\epsilon_{pqk}A_{j}B_{l}C_{m}=(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})A_{j}B_{l}C_{m}...

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LaTeX fonts for "epsilon" There are two "epsilons", one that looks like the set membership, and the other that looks like a backwards 3. I want the second one, but my latex software only outputs the first one. How do I change that?

I'm studying limits now (for the first time) and though have understood the intuitive concept of limit, I didn't get at all the epsilon-delta concept. What is epsilon and delta? What is x-2? I didn't get anything at all. So please explain me these in detail. Thanking you in advanced...

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I don't know how yor format this, so: x3 + x2 +x +1 The limit of that function = 0 as x approaches -1 What's the greatest value of delta when epsilon = 0.1? This is what I tried to do: |x3 + x2 + x + 1| < 0.1 -0.1 < x3 + x2 + x + 1 < 0.1 -1.1 < x3 + x2 + x < -0.9 In my...

How does one prove: limit xy = ab x-> a y -> b using the precise definition of a limit? My attempt: |xy-ab|<ϵ for: 0<|x-a|<δ/2 0<|y-b|<δ/2 it follows that: δ/2-a <x< δ/2+a δ/2-b <y< δ/2+b then: (δ/2-a)(δ/2-n) < xy < (δ/2+a)(δ/2+b) (δ^2/4-aδ/2-bδ/2 +ab)...

Okay for a simple finite limit: e.g. lim (3x) = 3 x->1 in the end I say: "Therefore for every |x - 3| < delta, there exists an epsilon such that |3x-3| < epsilon" Hence I can make delta really really small and the y bounds of epsilon will constrain the limit. So let's come to...

Suppose $\lim_n \frac{a_n -1}{a_n +1} = 0$. Prove that $\lim_n a_n = 1$. I am trying to do the algebra so that -a_n < ?? < a_n , but I am having trouble. Am I going about this correctly? I have also tried to solve each separate side of the inequality. I get a_n < (e+1)/(1-e), but this is...

This is a famous proof that utilizes a common notion. Theorem. Limits are unique. let n>N_1 such that blah blah blah is less than epsilon over 2, let n>N_2 such that blah blah blah is less than epsilon over 2. For n> max{N_1,N_2}, blah blah blah < blah = epsilon...

1. The problem statement. Give a complete and accurate \delta - \epsilon proof of the thereom: If f is differentiable at a, then f is continuous at a. 2. The attempt at a solution Known: \forall\epsilon>0, \exists\delta>0, \forall x, |x-a|<\delta \implies \left|\frac{f(x) - f(a)}{x-a}...

[SOLVED] Epsilon Delta Proof Does this limit proof make total sense? Given : "Show that \lim_{x \rightarrow 2} x^{2} = 4." My attempt at it :0<|x^{2}-4|<\epsilon which can also be written as 0<|(x-2)(x+2)|<\epsilon. 0<|x-2|<\delta where \delta > 0. It appears that \delta = \frac...

I can't get my head around the epsilon-delta definition of a limit. Unfortunately I don't have a teacher to ask (I'm teaching this to myself as a self interest) so this forum is my last resort -- google hasn't been kind to me. From what I've seen, I don't really understand how the definition...

I’m going to say from the beginning that I need to hand this problem in. I'm not looking for the answer, I think I already have it, just want a critical eye. I need someone to look over this problem and tell me if it's good. Not just if it's right but if it's perfect. I always get the...

Homework Statement Is this the right direction to prove Given that , prove that . Using the delta epsilon definition to prove that means that, for any arbitrary small there exists a where as: If we choose any constant for (x) called C, as long as C does not equal zero, the...

Homework Statement i have a couple of questions to anser and they start 'Give epsilon - delta proofs that the following functions are continuous at the indicated points.' im guessing its not going to be too hard but what is the name of this epsilon - delta proof so i can search for and...

Homework Statement Prove: \lim_{x \to a} \sqrt{x} = \sqrt{a}, if a>0 Homework Equations |f(x)-L| < \epsilon |x-a| < \delta The Attempt at a Solution |\sqrt{x}-\sqrt{a}| < \epsilon, when |x-a| < \delta |x - 2\sqrt{x}\sqrt{a}-a| < \epsilon^2, when |x-a| < \delta From here I...

Could someone please give me a walkthrough of the following question(and answer)?? I really can't understand it... lim x^2 = 9 x->3 if 0<|x-c|<delta then |f(x) - L|< epsilon so... x^2 - 9 = (x+3)(x-3) |x^2 - 9| = |x+3||x-3| Here's the problem.The book states: An...

I'm having trouble solving this question... If the epsilon ring of Uranus is 75 km wide, how long will it occult a star? Any help would be greatly appreciated.

All known natural materials have epsilon and mu > 0. However, there is an intense effort to manufacture so-called metamaterials, where both of these are <0. In such a case, given n = sqrt[epsilon x mu], one would take the negative root and n is still a real number, but n <0 why do we take the...

How do I write that given a set S, every epsilon neighborhood of infinity (in the complex plane) contains at least one point of S?

Is there a value for epsilon in the equation: k=1/4pi-epsilon?

consider Coulomb's law in medium. You find there epsilon. Consider the expression for the magnetic field generated by an electric current in the same medium (nonconducting, homogeneous and isotropic). Please let me know how do they transform when we consider them from two inertial reference...

Homework Statement Given that lim f(x)=L as x approaches a , prove that lim (x+f(x))=a+L as x approaches ahttps://www.physicsforums.com/attachments/9630. Your proof cannot assume that the limit of a sum of two functions is the sum of their individual limits. You must use the delta-epsilon...

Homework Statement find a number delta Homework Equations f(x) = 1/x | 1/x - 0.5 |<0.2 whenever | x - 2 | < delta The Attempt at a Solution how would you factor out a negative exponent? is this possible? i think i can get x out from under the 1/x with using negative...

I'm supposed to prove that lim x -> -2, x^2+3x+7 = 5 Here's what I have: |x^2 +3x+7 – 5| < ε |x+2| < δ |x^2+3x+2| -> |(x+2)(x+1)| < ε whenever |x+2| < δ |x+1||x+2| < δ |x+1| |x+1||x+2| < δ|x+1| < ε ε / |x+1| > δ , as x -> -2, |x+1| -> 3, therefore: ε / -1 > δ But...

Hi, Why is it, that when ever epsilon-delta proofs are done, once delta is found in terms of epsilon, it is reinputed through again? Is there any point to this really?

Can anyone help me on this question,Using \epsilon-\delta definition of a limit to show that \lim \frac {2x^3-y^3}{x^2+y^2} = 0 (x,y)\rightarrow(0,0)