What is Epsilon delta: Definition and 79 Discussions

Kappa Delta Epsilon (ΚΔΕ) is a professional fraternity for students in Education. It was organized on March 25, 1933.

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  1. A

    MHB Proof that lim loga_n/n = 0 in epsilon delta language

    Let $\,a>0\,,\,a\neq1\,$ be a real number. We can prove by using the continuity of $\ln n$ function that $\;\lim\limits_{n\to\infty}\dfrac{\log_an}n=0\;$ However, this problem appears in my problems book quite early right after the definition of $\epsilon$-language definition of limit of a...
  2. I

    Epsilon delta proof of the square root function

    Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
  3. Beelzedad

    I Does This Limit Exist?

    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...
  4. N

    I Exploring the Benefits of Epsilon-Delta Proofs in Mathematical Analysis

    I understand the concept of Epsilon-Delta proofs, but I can't understand why we have to do them. What's the advantage of using this proof over just showing that the limit from the function approaches from the left and right are the same?
  5. M

    Ε-δ proof: lim x->a f(x) = lim h->0 f(a + h)

    This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...
  6. J

    Help with Epsilon Delta Proof of Multivariable Limit

    Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...
  7. ChloeYip

    B Choosing Epsilon for a Limit Problem: A High School Student's Guide

    I have only encountered questions that f(x)-L that can be factorize to get a constant, and delta is epsilon divide that number, as a high school student. I have no idea how to choose a epsilon for this question. Thanks.
  8. Alpharup

    I Epsilon-Delta definition property.

    he definition of the limit of a function is as follows:[5] Let be a function defined on a subset https://upload.wikimedia.org/math/a/1/b/a1b67abab803e714098f3e69a33900da.png, let be a limit point of https://upload.wikimedia.org/math/f/6/2/f623e75af30e62bbd73d6df5b50bb7b5.png, and let be a...
  9. AutumnWater

    Epsilon Delta proof of a 2variable limit. Is my proof valid?

    Homework Statement Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0 Homework Equations sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)? The Attempt at a Solution This limit is true IFF for all values of epsilon > 0, there...
  10. AutumnWater

    Will epsilon delta test fail if curve changed direction?

    Will epsilon delta test fail if curve changed direction within the +/- delta of the limit point? Is there a scenario where no matter how small we pick delta to be, the frequency of the graph changing directions is always going to be a higher than delta's distance? In that scenario the limit...
  11. Avatrin

    Epsilon delta definition of limit

    I am struggling to properly understand the \varepsilon-\delta definition of limits. So, f(x) gets closer to L as x approaches a. That is okay. However, taking the leap from there to the \varepsilon-\delta definition is something I have never really been able to do. Why is the formulation we...
  12. F

    Where is the mistake in this epsilon delta proof?

    The question asks to proof that the limit given in incorrect by contradiction. I tried working using the estimation method and the weird thing is that I completed the proof and found that the supposedly "incorrect" limit gave a correct answer although it was supposed to give me a contradiction...
  13. K

    Epsilon delta limits if/then language

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > This is from the question list at the UC Davis Website epsilon delta exercise list. In the exercise list we have: Prove that Which concludes with: Thus, if , it follows that...
  14. W

    Understanding the Epsilon Delta Definition of a Limit

    Hi I'm new to limits and calculus in general. Our professor told us there existed some rigorous proof for a limit, but it was "beyond the scope of the course". All we needed to know about a limit was that (1)$$\lim_{x\to a} f(x)$$ is true iff when x approaches a from both directions p(x)...
  15. Dethrone

    MHB Limit 2^n/n using epsilon delta

    Using epsilon delta, prove $$\lim_{{n}\to{\infty}}\frac{2^n}{n!}=0$$ Doesn't seem too difficult, but I have forgotten how to do it. Obvious starting point is $\forall \epsilon >0$, $\exists N$ such that whenever $n>N,\left|\frac{2^n}{n!} \right|<\epsilon$.
  16. U

    Ross Elementary Analysis Epsilon Delta Proofs

    Does Ross's book teach and/or use Epsilon-delta proof techniques?
  17. U

    Calculus Spivak Calculus on Manifolds and Epsilon delta proofs

    I am currently having some issue understanding, what you may find trivial, epsilon-delta proofs. I have worked through Apostol Vol.1 and ran through Spivak and I found Apostol just uses neighborhoods in proofs instead of the epsilon-delta approach, while nesting neighborhoods is 'acceptable' I...
  18. Dethrone

    MHB Epsilon Delta Proof for 1/x

    $$\lim_{{x}\to{2}}\frac{1}{x}=\frac{1}{2}$$ 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...
  19. Dethrone

    MHB Epsilon Delta Proof Piecewise function

    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}...
  20. D

    Epsilon Delta Proof of a Limit

    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...
  21. LeibnizIsBetter

    MHB Epsilon delta proof of a two-variable limit using inequalities

    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 certain way...
  22. Seydlitz

    Variation of Epsilon Delta Proof

    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...
  23. S

    Real analysis epsilon delta problem

    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...
  24. Avatrin

    Epsilon delta continuity of 1/x at x=1

    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...
  25. H

    Why use stipulations in Epsilon Delta Proofs?

    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...
  26. PeteyCoco

    Does this epsilon delta limit proof check out?

    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...
  27. F

    Prove limits using epsilon delta definition

    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
  28. O

    Multivariable epsilon delta proofs

    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)) /...
  29. M

    Epsilon Delta Limit Definition

    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...
  30. C

    Proof of lim(1/x) x->0 by negating epsilon delta definition of limit

    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...
  31. T

    Epsilon delta proofs equaling a constant

    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!
  32. D

    Need help with epsilon delta proof of f(x)=x^4+(1/x) as x goes to 1

    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...
  33. D

    Epsilon delta proof that x^4 goes to a^4 as x goes to a

    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...
  34. F

    Solving Epsilon Delta Proof: lim 3 as x->6 & lim -1 as x->2

    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...
  35. H

    Epsilon Delta Definition of Limit

    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...
  36. A

    Solving Epsilon-Delta Proof: Homework Statement

    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...
  37. Ƒ

    Epsilon Delta Limit Solved

    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...
  38. K

    Solving Epsilon Delta Proof: |x^2 - 9| < 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 <...
  39. K

    Does anybody know a site where i can find many epsilon delta problems?

    I'm sick of still not getting this. I bombed the epsilon delta part of my mid term. A site where it gives many problems on epsilon delta and solutions would be amazing.
  40. N

    Need help proving a limit using epsilon delta definition.

    Need help proving lim(x)->(a) sqrt(x)=sqrt(a) using epsilon delta definition. Homework Statement Prove that the limit of \sqrt{x} is \sqrt{a} as x approaches a if a>0 Homework Equations in words By the epsilon delta definition we know that the distance between f(x) and the limit...
  41. J

    Beginner's Epsilon Delta Proof: Help needed

    Homework Statement This is my first delt/epsilon proof ever, so please understand if I seem ignorant. e=epsilon d = delta Let f(x) = 1/x for x>0 If e is any positive quantity, find a positive number d, which is such that: if 0 < |x-2| < d, then |f(x) - 1/2| < e Homework...
  42. J

    Epsilon Delta Limits: Finding \delta

    Homework Statement Suppose |f(x)-5|<0.1 when 0<x<5. Find all values \delta>0 such that |f(x)-5|<0.1 whenever 0<|x-2|<\delta Homework Equations The Attempt at a Solution I know that 0<|x-2|<\delta 2-\delta<x<2+\delta \delta=2 but how does this part of the equation help me find...
  43. W

    Proofs with epsilon delta (real analysis)

    Hello, I have stumbled upon a couple of proofs, but I can not seem to get an intuitive grasp on the what's and the whys in the steps of the proofs. Strictly logical I think I get it. Enough talk however. Number 1. "Let f be a continuous function on the real numbers. Then the set {x in R ...
  44. D

    Help on epsilon delta proof of discontinuity

    Homework Statement Prove the function f(x)= { 4 if x=0; x^2 otherwise is discontinuous at 0 using epsilon delta. Homework Equations definiton of discontinuity in this case: there exists an e>0 such that for all d>0 if |x-0|<d, |x^2-4|>e The Attempt at a Solution I'm confused...
  45. M

    Epsilon delta limit problem

    Homework Statement find an open interval about x0 on which the inequality lf(x)-Ll<\epsilon holds. Then give a value for \delta>0 such that for all x satisfying 0<lx-xol<\delta the inequality lf(x)-Ll<\epsilon holds f(x)=x2 L=3 x0=-2 \epsilon=0.5 Homework Equations The Attempt...
  46. F

    How Can I Prove Continuity Using Epsilon Delta Definition?

    Recently in adv calc we have been dealing with the epsilon delta definition for continuity, and my professor said that it is ok to assume that delta<1. I actually used this to show that x^4 satisfies the epsilon delta condition but I'm not quite sure why we can take delta<1. I am sure you guys...
  47. L

    How Do I Construct an Epsilon-Delta Proof for a Function Mapping R2 to R1?

    I have a problem on a take-home test, so I can't ask about the specific problem. So this is just going to be a general, how do I put stuff together problem. I have a function of x and y that maps R2 into R1. The limit as (x,y)->(0,0) is zero, and I've worked through the various paths already...
  48. M

    Understanding Epsilon Delta Limits Relationship in Calculus

    Homework Statement Is my understanding correct? As delta(y) and delta(x) approach X from points to the left and points to the right of X (x is what we wish to find the derivative of) then the x and y values of points to the left and right approach the x and y values of X. And as the...
  49. A

    Using Epsilon Delta to prove a limit

    Homework Statement prove that the lim as x goes to 4 of x^2 + x -11 = 9 This is the example used on Paul's Online Notes on limits in calculus which can be found here http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfLimit.aspx (I really like this resource.) Homework Equations Paul...