Epsilon Definition and 215 Threads

Hi, Suppose you want to prove |x - a||x + a| < \epsilon You know |x - a| < (2|a| + 1) You need to prove |x + a| < \frac{\epsilon}{2|a| + 1} So that |x - a||x + a| < \epsilon Why does Michael Spivak do this: He says you have to prove --> |x + a| < min(1, \frac{\epsilon}{2|a| + 1}) in...

$$\lim_{{x}\to{2}}\frac{1}{x}=\frac{1}{2}$$ Here is what I have so far: For all $\delta >0$, there exists an $x$ such that $0<|x-2|<\delta $, $|\frac{1}{x}-\frac{1}{2}<\epsilon$ Cut to the chase: $$\frac{|x-2|}{|2x|}<\epsilon$$ I need to bound $\frac{1}{|2x|}$ somehow, and represent it with...

https://answers.yahoo.com/question/index?qid=20130915100124AAK4JAQ I do not understand how they got: "1 = |(1 plus d/2 - L) - (d/2 - L)| <= |1 plus d/2 - L| plus |d/2 - L| < 1/4 plus 1/4 = 1/2, " Shouldn't it be $|(1+ \frac{\delta}{2} -L) + (\frac{\delta}{2} -L)|$, not $|(1+ \frac{\delta}{2}...

Verify, by a geometric argument, that the largest possible choice of $\delta$ for showing that $\lim_{{x}\to{3}}x^2=9$ is $\delta = \sqrt{9+\epsilon}-3$ I have no clue, hints?

Prove that $\lim_{{x}\to{0}}\frac{1}{x^4}=\infty$, given a $M>0$ So we need to prove that $f(x) > M$: $\frac{1}{x^4}>M$, $\frac{1}{M}>x^4$, $\frac{1}{M^{1/4}}>|x|$ Is that right so far? Is the absolute values necessary in my last statement?

can someone explain it?

Suppose we are given the function $y=2+\frac{1}{x^2}$. Prove that given $x>\frac{1}{(\epsilon)^{1/2}}$, where $\epsilon > 0$, then $2- \epsilon < y < 2 + \epsilon$. So the first part is easy: $$x>\frac{1}{(\epsilon)^{1/2}}$$ $$x^2>\frac{1}{\epsilon}$$ $$\frac{1}{x^2}<\epsilon$$...

Homework Statement Prove that ## lim_{x\implies 1} \frac{2}{x-3} = -1 ## Use delta-epsilon. The Attempt at a Solution Proof strategy: ## | { \frac{ 2}{x-3} +1 } | < \epsilon #### \frac{x-1}{x-3} < \epsilon ## , since delta have to be a function of epsilon alone and not include x. I...

these proofs are always confusing but here's my take on it.. since $x\rightarrow +\infty$ we don't need absolute values and since $ \displaystyle \frac{1}{10^2}=0.01 $ then we could use $N=10$ letting $L=0$ since it is a horz asymptote then we have $ \displaystyle...

So I made a little application to show the steps of approximating a squareroot using the Babylonian Method: http://jaminweb.com/squarerootCalc.html It's not working all the time. When I do the squareroot of 25 starting with an initial guess of 3.1, it works: Applying...

Homework Statement Prove the following sequence {an} converges to L=1/2 an = n2/(2n2+n-1) The Attempt at a Solution Given ε>0 we can determine an N∈N so that |an - L|<ε for n≥N. We have: |an-L|=|(n2/(2n2+n-1)-(1/2)| = |(-n+1)/(2(2n-1)(n+1))| I'm not sure what to do once I get to this...

Prove Lim x^2=9. With the epsilon/delta definition of a limit. x->3 My work so far. For every ε>0 there is a δ>0 such that if 0<|x-3|<δ , Then |x^2-9|<ε so, |(x-3)(x+3)|<ε |x-3|* |x+3|∠ε what do I do from here? My book is not very clear (Stewart Calculus 7ed)...

Find the limit L. Then use the epsilon-delta definition to prove that the limit is L. $\sqrt(x)$ as x approaches 9 I figure out the first part of the question. the Answer is three. Yet I have some difficulty to answer the second part of the question.Thank you Cbarker11

Homework Statement Use a graphing calculator to find \delta when 0<|x - \pi/2|<\delta and |sin(x) - 1|<0.2 Homework Equations I don't think there are any other than the format of the previous information: 0<|x-a|<\delta and |f(x)-L|<\epsilon The Attempt at a Solution Okay...

I am trying to check whether lim h→0 (R(h)/||h||) =0 or not. I am working in ℝ2. h=h1e1+h2e2** => ||h||=(h1^2+h2^2)^1/2 I am using the definition that (R(h)/||h||)<ε * whenever 0<|h|<δ for all h. Example 1 (R(h)/||h||)=h1h2/(h1^2+h2^2)^3/2 I can see that the denominator dominates...

Hi, I have been trying to find out the derivation of Gauss's law but can't seem to find any derivations. May I know how the differential form of Gauss's law is derived and where does the epsilon come from? Does it have to do with the displacement field definition?

Hey there, I'm new to this forum. Today I thought I would brush up on my calculus. I would just like to know if my method is correct. Is there an easier way to prove this ? By the way, it's my first time using LaTeX, so bear with me. I am trying to prove the following : \lim_{x\rightarrow...

I seem to be having trouble with multivariable epsilon-delta limit proofs. I don't have a very good intuition for how $$\epsilon$$ relates to $$\delta$$. For example: $$Prove \lim_{(x,y) \to (0,0)}\frac{2xy^2}{x^2+y^2} = 0$$ There are probably many ways to do this, but my teacher does it a...

Hello, I am a physics undergrad with no experience in analysis and I have not had to do any proofs before. I am taking a class in complex analysis as an elective. I have been doing well in the course so far and improving on my rigour in proofs and such. On to my question: As...

Homework Statement I need to prove that 1/n has a limit of zero using the following definition: The statement that the point sequence p1, p2, . . . converges to the point x means that if S is an open interval containing x then there is a positive integer N such that if n is a positive...

Hi. I'm studying numerical methods so I found this subforum the most correct for this question. The machine epsilon for a computer is defined as the least number e such that 1 + e is different to 1. I just wonder, why 1 + e? And not 2 + e for instance? Thanks!

Homework Statement Prove that if ##\left |x-x_{0} \right | < \frac{\varepsilon }{2}## and ##\left |y-y_{0} \right | < \frac{\varepsilon }{2}## then ##|(x+y)-(x_0+y_0)| < \varepsilon ## and ##|(x-y)-(x_0-y_0)| < \varepsilon ##Homework Equations Postulate and proof with real numbers as well...

I've been reading through Spivak's calculus, and the problem is the answer key i have a hold of is for a different edition so it often doesn't answer the correct questions. Anyways, here they are: Chapter 5 problem 10 b. Prove that lim x-> 0 f(x) = lim x-> a f(x-a) c. Prove that lim...

This isn't really homework; It's just something that has been bothering me ever since I first learned calculus because I suck at epsilon-delta proofs. Homework Statement Show that 1/x is continuous at x=1 Homework Equations If |x-a|<δ Then |f(x)-f(a)|<ε The Attempt at a Solution...

Homework Statement When constructing an Epsilon Delta proof, why do we need to make a stipulation? For example, in most proofs for limits of quadratic functions, it is stipulated, for example, that δ≤1. Why is this needed anyway? This is my thought process for a quadratic: Prove that lim(x...

Homework Statement Suppose x,y \in X which is a normed linear space and x\neq y . Prove that \exists r>0 such that B(x,r) \cap B(y,r)=∅ Homework Equations Epsilon Ball B(x,r)={z \in X:||x-z||<r} The Attempt at a Solution So my attempt here is via contradiction and its not...

I started learning how to do these things today and boy, they take some interesting logic. Anyway, here's my attempt at one: prove that the limit as (x,y) → (0,0) of [(x^2)(siny)^2]/(x^2 + 2y^2) exists Here's what I did: 0<√(x^2 + y^2) < δ, |[(x^2)(siny)^2]/(x^2 + 2y^2) - 0| < ε...

Homework Statement http://store2.up-00.com/Sep12/JB498124.jpg 2. The attempt at a solution No attempts because i can't understand how to solve it

Homework Statement lim (x,y) -> (0,0) xy/sqrt(x^2+y^2) = 0 The Attempt at a Solution my understanding of my actual goal here is kind of poor given ε>0 there exist ∂>0 s.t. 0 < sqrt(x^2 + y^2) < ∂ then 0<|f(x,y) - L| < ε | xy/sqrt(x^2 + y^2) - 0 | < ε (xy * sqrt(x^2 + y^2)) /...

Homework Statement Prove lim x--> -1 1/(sqrt((x^2)+1) using epsilon, delta definition of a limit Homework Equations The Attempt at a Solution I know that the limit =(sqrt(2))/2 And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if...

Hi, I am wondering how one would go about an ε, N proof for a recursively defined sequence. Can anyone direct me to some reading or would like to provide insights of their own? This isn't for a homework problem... just general curiosity which I could not satisfy via search! Thank you.

Homework Statement I want to show that \lim_{x \rightarrow 0}\frac{1}{x} does not exist by negating epsilon-delta definition of limit. Homework Equations The Attempt at a Solution We say limit exists when: \forall \epsilon > 0, \exists \delta > 0 : \forall x(0< \left| x\right| < \delta...

Homework Statement I'm wondering why we can't use less than or equal to for the formal definition of the limit of a function: Homework Equations lim x→y f(x)=L iff For all ε>0 exists δ>0 such that abs(x-y)<δ implies abs(f(x) - L)<ε Why not: lim x→y f(x)=L iff For all ε>0 exists...

Homework Statement Prove that (AxB) is perpendicular to A *We know that it is in the definition but this requires an actual proof. This is what I did on the exam because it was quicker than writing out the vectors and crossing and dotting them. Homework Equations X dot Y = 0 when...

proving the "contracted epsilon" identity in the wikipedia page for the Levi Civita symbol, they have a definition of the product of 2 permutation symbols as: ε_{ijk}ε_{lmn} = δ_{il}(δ_{jm}δ_{kn} - δ_{jn}δ_{km}) - δ_{im}(δ_{jl}δ_{kn} - δ_{jn}δ_{kl}) + δ_{in}(δ_{jl}δ_{km} - δ_{jm}δ_{kl}) and...

Homework Statement Lim x→a of f(x) = c (Where c is a constant) Homework Equations The Attempt at a Solution I have no idea. I am able to do these if I can manipulate fx-L to equal x-a but I am having trouble with this one. Please help me!

Homework Statement Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ. f(x)=x^{4}+\frac{1}{x}, a=1. Homework Equations I claim that \lim\limits_{x\rightarrow 1}x^{4}+\frac{1}{x}=2. The...

Homework Statement Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ. f(x)=x^{2}, arbitrary a.Homework Equations I will incorporate the triangle inequality in this proof.The Attempt at a Solution We...

Homework Statement lim 3 as x->6 lim -1 as x->2 Homework Equations In the first weeks of a calculus class and doing these epsilon delta proofs. As i am looking at two of the problems i have been assigned: Lim 3 as x->6 Lim -1 as x->2 The Attempt at a Solution...

Use the delta-epsilon definition to prove f(x,y) = y/(1+x2) is continuous at (0,0)Attempts:So I'm doing some work and my main issue is finding a bound for the denominator of 1+x2: So work wise I have something looking like: \delta/(|1| + |x2| ). How could I found a good bound?

So far, all I understand is that the definition proves that there's a value of f(x,y) as f(x,y) approaches (x0,y0) which is sufficiently close to but not exactly the value at f(x0,y0). I am probably completely off... but I just don't understand the purpose of proving this. I also don't...

Really couldn't catch the concept on epsilon and delta in limits. Let ∂x=x2 - x1 In finding a gradient the value ∂y is taken at certain value. But in finding area using integral, the ∂y is seen to taken as zero. F(x2)=F(x1) Maybe one multiplication and the other is division.

Homework Statement I just want to make sure I include all the steps in doing this: lim (6x-7) = 11 x->3 Homework Equations The Attempt at a Solution given ε>0, we need to find a δ>0, such that 0< lx-3l < δ then 0 < l (6x-7)-11 l < ε To prove this I need to make 0 < l...

Simple enough - I'm just trying to find the smallest positive real number, ε, such that 1 + ε ≠ 1 in MATLAB (double precision). So the value 'eps' in MATLAB is actually not quite defined this way, and using this program min = 0; max = 1; test = 1; while test~=(min+max)/2 test =...

For part A, (described here: http://www.cramster.com/solution/solution/1157440) I don't understand why they say |x-2| < 1 and why \delta = min{1,ε/5} In case you can't view the page: lim x2+2x-5 = 3, x \rightarrow 2 Let ε > 0 and L = 3. |x2 + 2x -5 -3| < ε |x2 + 2x - 8| < ε |x+4||x-2| < ε...

Homework Statement if |x-3| < ε/7 and 0 < x ≤ 7 prove that |x^2 - 9| < ε Homework Equations The Attempt at a Solution So ths is what I did so far. |x+3|*|x-3| < ε (factored out the |x^2 - 9|) |x+3|*|x-3| < |x+3|* ε/7 < ε (used the fact that |x-3| < ε/7) |x+3|* ε/7 *7 <...

In all the problems I have done so far, the limit was already given. So the goal is to utilize the theorem to see whether the limit really holds. But what's the point? If we already know how to find the limit, why must we go through a process of ingenuity algebra to tell ourselves, "okay it...

Homework Statement I already have the solutions, but I am not sure what the solutions are trying to say. http://img194.imageshack.us/img194/2595/unledlvc.jpg So in I don't understand this, we have n > \frac{1}{\epsilon} and If (and I am guessing we really want this to...

Homework Statement Prove, using the formal definition of limits: If http://rogercortesi.com/eqn/tempimagedir/eqn4201.png and c>0, then [PLAIN]http://rogercortesi.com/eqn/tempimagedir/eqn4201.png (add the constant beside f(x) here, I couldn't get the equation generator to cooperate)...

Homework Statement Prove, using the formal definition of limits: If lim (x->inf) g(x) = inf and g(x) leq f(x) for x->a, then lim (x->a) f(x)=inf. leq = less than or equal to. Homework Equations The Attempt at a Solution Honestly, I'm not even sure where to start on this one. Anyone bored...