What is Epsilon: Definition and 221 Discussions

Epsilon (UK: , US: ; uppercase Ε, lowercase ε or lunate ϵ; Greek: έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel /e/. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.
The name of the letter was originally εἶ (Ancient Greek: [êː]), but the name was changed to ἒ ψιλόν (e psilon "simple e") in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon.
The uppercase form of epsilon looks identical to Latin E but has its own code point in Unicode: U+0395 Ε GREEK CAPITAL LETTER EPSILON. The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed number "3" and is encoded U+03B5 ε GREEK SMALL LETTER EPSILON. The other, also known as lunate or uncial epsilon and inherited from earlier uncial writing, looks like a semicircle crossed by a horizontal bar: it is encoded U+03F5 ϵ GREEK LUNATE EPSILON SYMBOL. While in normal typography these are just alternative font variants, they may have different meanings as mathematical symbols: computer systems therefore offer distinct encodings for them. In TeX, \epsilon (


{\displaystyle \epsilon \!}
) denotes the lunate form, while \varepsilon (


{\displaystyle \varepsilon \!}
) denotes the reversed-3 form.
There is also a 'Latin epsilon', ɛ or "open e", which looks similar to the Greek lowercase epsilon. It is encoded in Unicode as U+025B ɛ LATIN SMALL LETTER OPEN E and U+0190 Ɛ LATIN CAPITAL LETTER OPEN E and is used as an IPA phonetic symbol. The lunate or uncial epsilon provided inspiration for the euro sign, €.The lunate epsilon, ϵ, is not to be confused with the set membership symbol ∈; nor should the Latin uppercase epsilon, Ɛ, be confused with the Greek uppercase Σ (sigma). The symbol

{\displaystyle \in }
, first used in set theory and logic by Giuseppe Peano and now used in mathematics in general for set membership ("belongs to") evolved from the letter epsilon, since the symbol was originally used as an abbreviation for the Latin word "est". In addition, mathematicians often read the symbol ∈ as "element of", as in "1 is an element of the natural numbers" for



{\displaystyle 1\in \mathbb {N} }
, for example. As late as 1960, ε itself was used for set membership, while its negation "does not belong to" (now ∉) was denoted by ε' (epsilon prime). Only gradually did a fully separate, stylized symbol take the place of epsilon in this role. In a related context, Peano also introduced the use of a backwards epsilon, ϶, for the phrase "such that", although the abbreviation "s.t." is occasionally used in place of ϶ in informal cardinals.

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  1. P

    A $\phi^4$ in $4 - \epsilon$ dimension renormalization beta function

    Hi all, I am currently studying renormalization group and beta functions. Since I'm not in school there is no one to fix my mis-understandings if any, so I'd really appreciate some feedback. PART I: I wrote this short summary of what I understand of the beta function: Is this reasoning...
  2. 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...
  3. E

    A Exploring Epsilon Simultaneity: Advantages and Applications

    What is the advantage of considering the generalised simultaneity criterion ##t = (1-\epsilon)t_1 + \epsilon t_2## for ##\epsilon## between ##0## and ##1##? How does varying the parameter ##\epsilon## help to elucidate the structure of the special theory? I think the surfaces of simultaneity are...
  4. A

    I Riddle: Where is Epsilon in Bohr Model of Atom?

    Hi curious people Today i had a lecture about bohr model of the atom .we know that the total energy of electron in orbit is minus e squared divided by 8 pi times epsilon times R of the orbit . this is written in my book and i agree with this .But when the author writes a formula for an emitted...
  5. karush

    MHB S8.1.7.6 delta epsilon proof

    Find a graph to a number $\delta$ such that $$\textit{if } |x-1|<\delta \textit{ then } \left|\dfrac{2x}{x^2+4}-0.4\right|<0.1 $$ ok I always had a very hard time doing these I did look at some examples but still ? did a ibispaint drawing to start basically it looks like we are finding the...
  6. 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| <...
  7. N

    I Why use Epsilon Delta proofs?

    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?
  8. reese houseknecht

    Calculating how to make a homemade capacitor

    Ok, so I am am trying to make a homemade capacitor that is 4700pf and 15kv. polyester at 125 micrometers thick can withstand 15kv so were good. here is the equation i used C=ε0 K A / D Where C = capacitance, ε0 is epsilons constant, K = dielectric constant, A = area of aluminum foil, and D is...
  9. D

    LaTeX Sample problems; Simple Limit (Epsilon - Delta proofs) Latex code included

    Remember to use the appropriate packages; these are in similar post if a mod wants to add the link if you choose to use Latex. Here is the PDF \begin{document} \begin{center} {\LARGE Epsilon-Delta Proofs \\[0.25em] Practice} \\[1em] {\large Just for practice, don't use Google to cheat!}...
  10. Mr Davis 97

    Proving the Triangle Inequality: ##|a-b| < \epsilon##

    Homework Statement If ##\forall \epsilon > 0 ## it follows that ##|a-b| < \epsilon##, then ##a=b##. Homework EquationsThe Attempt at a Solution Proof by contraposition. Suppose that ##a \neq b##. We need to show that ##\exists \epsilon > 0## such that ##|a-b| \ge \epsilon##. Well, let...
  11. Cosmophile

    Finding Delta Given Epsilon with a Quadratic Function

    Hey, everyone! I'm helping a friend through his calculus course and we've come across something that has stumped me (see: the title). When I learned calculus, our treatment of the epsilon-delta definition of the limit was, at best, brief. Anyway, here is the problem: Given ##\lim_{x \rightarrow...
  12. AdrianMachin

    Multiplying epsilon naught by a length quantity

    Homework Statement Note that this formula (##C=4 \pi \epsilon_0 R##) and the others we have derived for capacitance involve the constant multiplied by a quantity that has the dimensions of a length. Homework Equations ##\epsilon_0## has the following units in SI: $$\frac {C^2} {N \cdot m^2}$$...
  13. K

    Continuity: Epsilon & Delta Homework

    Homework Statement Homework Equations Continuity: $$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$ $$\delta=\delta(c,\epsilon)$$ The Attempt at a Solution $$\vert f(x)-f(c) \vert <\frac{1}{2}f(c)~\Rightarrow~\vert x-c \vert < \delta_1$$ So i have this δ1 but what...
  14. K

    Epsilon and delta of x^3

    Homework Statement find a δ for a given ε for f(x)=x3 around c=5: $$\vert x-5\vert<\delta~\Rightarrow~\vert x^3-5^3 \vert < \epsilon$$ Homework Equations Continuity: $$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$ $$\delta=\delta(c,\epsilon)$$ The Attempt at a...
  15. Mr Real

    I Constant raised to complex numbers

    It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...
  16. Bunny-chan

    Epsilon distance between two terms

    Homework Statement How close is x to x_0 (x_0 \neq 0) so that 2. Homework Equations The Attempt at a Solution I tried to use absolute value properties:- \epsilon \lt \frac{\sqrt{x_0^2+1}}{x_0^3} - \frac{\sqrt{x^2+1}}{x^3} \lt \epsilonBy adding in the three sides, we...
  17. J

    MHB Proving an Integral with a Direct Proof & Epsilon Argument

    Okay, these are my last questions and then I'll get out of your hair for a while. For 1, I have already done a proof by contradiction, but I'm supposed to also do a direct proof. Seems like it should be simple? For 2, this seems obvious because it's the definition of an integral. My delta is...
  18. X

    Find the induced electromotive force (epsilon)

    it is something like this but instead of the rectalve contour iswith the angle 37 degree!https://www.physicsforums.com/attachments/114624 1. Homework Statement the speed of the slide is 0.16 m/s the length of the slide is 0.14 m the angle of the contour is 37 degree Homework Equations...
  19. S

    I Epsilon in the limit definition

    in the limit definition of a sequence, why does epsilon has to be greater than 0 and not greater or equal to 0? thanks in advance.
  20. FallenApple

    I How can I prove that a sequence converges to a limit for every epsilon?

    Ok, I'm reviewing my real analysis and found a point of confusion. So for a sequence S(n) to converge to S, for every e>0 there exists N such that n>N implies |S(n)-S|<e. Ok that makes sense. However when I backsolve and find N as a function of epsilon, and then formally proving forwards, I'm...
  21. Austin Chang

    I Delta Epsilon Limit Problem

    Q's Let f,g ℝ→ℝ. Suppose that g is bounded. This means that its image is bounded or in other words there exists a positive real number B s.t. |g(x)| ≤ B ∀ x. Prove that if lim x→c f(x) = 0, then lim x→c f(x)g(x) = 0. Work. See the picture. I am really confused I can't seem to understand the idea...
  22. M

    Prove a limit using epsilon-delta definition

    Homework Statement Proof that: ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2}## using the epsilon-delta definition of a limit. (Problem below) Homework Equations ##\lim_{x \to 1} \frac{1}{1+x} = \frac{1}{2} \iff \forall ε>0, \existsδ>0, \forall x ∈ \mathbb{R}\backslash\{-1\}: 0 < |x-1| < δ...
  23. H

    General pathline of a particle x at point epsilon at time ta

    Homework Statement Show that the path line of a particle at point x currently, and point ξ at time τ is given by ξ(τ) = x + (τ-t)Lx Homework Equations Pathline is solution to dx/dt = u x(t)|t=τ = X L is the velocity gradient and is a 2nd order tensor Lij = dui/dxj The Attempt at a...
  24. 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...
  25. P

    I What is the opposite of epsilon

    Is there a notation for the opposite of epsilon (infinitesimal) in the way that infinity is the opposite of zero?
  26. I

    Given Epsilon > 0, come up with a delta? Limit problem.

    Lim as x approaches 4 of 1/x = 1/4 Given epsilon > 0, come up with a delta, d?Limits have been introduced. So far my instructor has had us make tables to see what value x was approaching. Although I don't understand exactly how limits are EVALUATED (different from looking at a chart & saying...
  27. 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...
  28. 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...
  29. R

    Constant on [epsilon, 2epsilon] implies f'(0)=K

    Homework Statement Warning: I realize the title is misleading... the function itself isn't what's constant. Mod note: Edited to fix the LaTeX If ##f## is a continuous at 0 such that ##\lim_{x \to 0}\frac{f(x)-f(g(x))}{g(x)}=M##, where ##g(x)\to 0## as ##x\to 0## does this generally mean that...
  30. 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...
  31. B

    Contracted Epsilon Identity

    Hi, I am confused about how I arrive at the contracted epsilon identity. \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} 1. Homework Statement Show that \epsilon_{ijk} \epsilon_{imn} = \delta_{jm} \delta_{kn} - \delta_{jn} \delta_{km} Homework EquationsThe...
  32. T

    MHB Trouble with Delta Epsilon Problem

    I understand and can do part (a) of the problem but when it gets to part (b) I'm lost, how do I find/explain why there is no value of delta > 0 that satisfies the the problem.
  33. 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...
  34. B3NR4Y

    Epsilon- N Proof a Sequence Diverges

    Homework Statement 1. Let ##x_n = \frac{n^2 - n}{n} ## does ##x_n## converge or diverge? 2. Let ##x_n = \frac{(-1)^n +1}{n} ## does ##x_n## converge or diverge Homework Equations A sequence converges if ##\forall \epsilon > 0##, ##\exists N \in \mathbb{N} ## such that ##n\geq N ## implies ##...
  35. A

    Why can we define μ naught?

    Hello all, While I understand the significance of natural units, I am wondering why, in SI units, we are able to assign μ0 an exact value. The speed of light is experimentally determined in m/s, and given the relationship derived from Maxwell's equations, we know that c^2 = 1/√(ε0μ0). Thus by...
  36. C

    Using the precise definition of a limit (epsilon & delta)

    I'm trying to practise, precise definition of a limit (epsilon & delta) Just to check I'm along the right lines here's a previous question to the one I'm stuck on If epsilon > 0 then there is delta >0 ... All that introduction stuff, then Lim x-> 2 (3x-1) =5 Hence |x-2| < delta then |3x - 6|...
  37. E

    ##\left|L(f_{n}, P) - L(f,P)\right| < \epsilon##

    Homework Statement Suppose that ##f_{n} \rightarrow f## uniformly on [a,b] and that each ##f_{n}## is integrable on [a,b]. Show that given ##\epsilon > 0##, there exists a partition ##P## and a natural number ##N## such that ##\left|L(f_{n}, P) - L(f,P)\right| < \epsilon##. Homework...
  38. 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...
  39. 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)...
  40. Dethrone

    MHB Proof of $\lim_{{n}\to{\infty}}a_n=\lim_{{n}\to{\infty}}b_n$ with $\epsilon-N$

    Let ${a_n}$ and ${b_n}$ be two convergent sequences such that $\lim_{{n}\to{\infty}}|a_n-b_n|=0$. Give an $\epsilon-N$ proof to show that $\lim_{{n}\to{\infty}}a_n=\lim_{{n}\to{\infty}}b_n$. Right now, I've just said without loss of generality, assume that $a_n>b_n$ for all $n>N$, then since...
  41. 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 $n>N,\left|\frac{2^n}{n!} \right|<\epsilon$.
  42. N

    Limits: 9-x as x Approaches 4 = 5

    lim(9-x) as x->4 = 5 I thought I was supposed to do this: 9-4=5 5=5 But apparently I was supposed to use delta and epsilon? I'm not sure how to find either of these. I know you find epsilon first but I'm really confused so if anyone knows just HOW to find it, that would be extremely helpful...
  43. N

    Delta and Epsilon Proofs

    Given a function f(x), a point x0, and a positive number E (epsilon), write the limit then find delta>0 such that for all x 0< |x-x0| < delta -> |f(x)-L| < E f(x) = 3-2x, x0=3, E=.02 Here is my attempt: Lim (3-2x) as x->3 = -3 -.02 < |3-2x - 3| <.02 -.02 < |-2x| < .02 .01 > x > -.01 -2.99 > x-3...
  44. N

    When people help they are not allowed to include answers?

    The website rules say that when people help they are not allowed to include answers? But how am I supposed to check my answers... anyone else have this problem?
  45. M

    Proving lim_{x->\frac{1}{10}}\frac{1}{x}=10

    Homework Statement Prove lim_{x->\frac{1}{10}}\frac{1}{x}=10 Homework Equations |f(x)-L|<epsilon, |x-a|<delta The Attempt at a Solution I need to go from 1/x to x, so I applied an initial condition of delta<1/20 \frac{-1}{20}<x-\frac{1}{10}<\frac{1}{20} \frac{1}{10}<x<\frac{3}{10}...
  46. C

    How can I find the distance between Epsilon Indi and Tau Ceti?

    Hi, I wonder if anyone can either answer the following or point me in the right direction for the answer? How far apart are Epsilon Indi and Tau Ceti? Thanks in advance. Chinspinner
  47. U

    Ross Elementary Analysis Epsilon Delta Proofs

    Does Ross's book teach and/or use Epsilon-delta proof techniques?
  48. 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...
  49. C

    Discovering Earth's View from Epsilon Indi: Answers by Chinspinner

    Hi, I wonder if anyone can answer the following: - What constellation would our sun appear in if viewed from Epsilon Indi? Thanks in advance Chinspinner