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Questions tagged [epsilon-delta]

For questions regarding $\varepsilon$-$\delta$ definitions of limits, continuity of functions and $\varepsilon-N$ definition of limit of sequences.

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Can someone help me find the hole in my $\epsilon$-$\delta$ argument? Hint needed (Spivak Chapter 5 Problem 20)

If $f(x)=x$ for rational $x$, and $f(x)=-x$ for irrational $x$,then $\lim_{x \to a} f(x)$ does not exist for any $a\neq 0$ Firstly we write down our epsilon delta argument $$0<|x-a|<\delta\...
Edward Falls's user avatar
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Does the value of a limit change if a choose a different kind of neighborhood?

Let $f\colon\Omega\to\mathbb R$ be a continuous function over an interval $\Omega\subseteq\mathbb R$. Let $x_0\in\Omega\setminus\partial\Omega$. By the usual definition of limit, we can say that $$\...
Elvis's user avatar
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3 answers
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Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$

We need to show that for every $\varepsilon >0$ there exists a $\delta >0$ such that $$\left\lvert\frac{x^3+y^3}{x^2+y^2}-\frac{7}{5}\right\rvert<\varepsilon $$ whenever $$0<\sqrt{(x+1)^2+(...
Afzal Ansari's user avatar
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Is my proof rigorous enough? Spivak, Prove:$\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$ (Chapter 5, Problem 10) [closed]

I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect). Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$. (...
Edward Falls's user avatar
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2 answers
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Trouble Understanding Difference in Epsilon-Delta Arguments: Why One Works But The Other Fails (Spivak Calculus Problem 5-10c)

In the problems for the limits chapter (5) of Spivak's Calculus, we are asked to prove: $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)$. The relevant to the question proof alternative is: ...
Stephen Premel's user avatar
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Help understanding Spivak's solution, and a verification of my proof. Spivak Chapter 5, Question 3(vi)

Find a $\delta$ such that if $0<|x-1|<\delta$ then $|\sqrt{x}-1|\lt \epsilon$ My solution: $|\sqrt{x}-1|\cdot |\sqrt{x}+1|=|x-1|$ $ \epsilon>|\sqrt{x}-1|\ge |\sqrt{x}|-1$ which implies $\...
Edward Falls's user avatar
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1 answer
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Understanding epsilon-delta proof regarding dense sets

I'm currently self-studying using Spivak's "Calculus" and I wanted to check on my understanding regarding an epsilon-delta proof for dense sets. The first problem was the following: If $f$ ...
Aryaan's user avatar
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Using Graph of $\frac{1}{x}$ to Find $\delta$ [closed]

Hello, This is a question I was assigned for homework and I am struggling to find a correct answer. The question states "Use the graph of $f(x)=\frac{1}{x}$ below to find a number δ such that $|f(...
Squishy698's user avatar
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If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following

If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following: $\lim_{x\to a}[f(x)+g(x)]=\infty$ $\lim_{x\to a}[f(x)g(x)]=\infty $ if $c>0$ $\lim_{x\to a}...
EpicFaceInc100's user avatar
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Finding $\delta$ for convergence of $f(x)=x^3+2x^2-3x-1$ at its approximate root.

I've had some exposure to elementary analysis and I am currently going through a problem in numerical analysis involving finding roots using Newton's method. The algorithm has convergence issues which ...
Mario Figueroa's user avatar
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2 answers
198 views

Proving that a function with only removable discontinuities can be made continuous

I'm working with Spivak's "Calculus" and was doing the following problem: Let $f$ be a function with the property that every discontinuity is a removable discontinuity. This means that $\...
Aryaan's user avatar
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The continuous functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x) = g(x)$ for all $x \in \mathbb{Q}$. [duplicate]

(a) Using the definition of continuity, prove that $f(x) = g(x)$ for all $x \in \mathbb{R}$. (b) Use sequential criteria of continuity to redo the problem. I was able to do the part (b) of this ...
Nicholas Gray's user avatar
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On continuous functions and convergent sequence

The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following. (a) If $\{x_n\}$ is convergent, then $\{f(x_n)\...
Nicholas Gray's user avatar
1 vote
1 answer
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Conditions equivalent to the ε-N definition

I want to know the condition equivalent to the ε-N definition. I think TFAE. But, I can't prove [2. ⟹ 1.]. $(a_n)$ is a sequence such that $\forall n\in \mathbb{N},a_n\in \mathbb{R}$,and $I\subset \...
nat-cat's user avatar
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Prove using epsilon - delta

let $f(x) = \dfrac {x-\sqrt x} {5x-4}$ prove using $\epsilon-\delta$ that $\lim_{x\to1} f(x) = 0$ I got it to - let $\epsilon >0$. choose $\delta = min(0.1 , \dfrac \epsilon 2)$ Than for all $\...
Id.d's user avatar
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Proof that $\lim_{x\to 0} f(x) = \lim_{x\to 0}f(x^3)$ [duplicate]

I wanted to prove the following theorem and was wondering if my line of reasoning was correct: Theorem: $\lim_{x\to 0}f(x) = \lim_{x\to 0}f(x^3)$ Proof: Suppose $\lim_{x\to 0}f(x)$ exists and equals $...
Aryaan's user avatar
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Verification of an epsilon-delta proof

I'm following Michael Spivak's "Calculus". Here's the question: $$0 < |x - 2| < \sin^2\left(\frac{\epsilon^2}{9}\right) + \epsilon\implies |f(x) - 2| < \epsilon$$ $$0 < |x-2| <...
Aryaan's user avatar
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Alternate epsilon-delta proof of the limit of the reciprocal function

I'm trying to prove the following using the epsilon-delta method: $$\lim_{x\to 1}\frac{1}{x} = 1$$ Prior to reading the fantastic answer by @Clarinetist in this question, I tried writing the proof ...
Aryaan's user avatar
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2 answers
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Find the limit and give the strict $\epsilon - \delta$ proof that: $\lim_{x\to 1^{+}}$ $x\lfloor x \rfloor$

This is from Advance Calculus Theory and Practice By John S. Petrovic. The book has the answer in the back but I need some clarity on $\delta$. Answer: The limit is 1. Let $\epsilon > 0$, and ...
Cat's user avatar
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Doubt about the Epsilon-Delta Definition of the limit of a function at $a$ point.

My math book states that for a limit to exist at a point $a$ of a function $f$, $f(a)$, must be equal to the limiting values from each side of the point. However, how does the epsilon-delta definition ...
Hudson Williams's user avatar
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Proof that the extremum of a function is a critical point [duplicate]

I'm following a proof given by Spivak in his textbook "Calculus" and one of the steps seems slightly unjustified to me. Firstly, here's the theorem: If $f$ is a function defined on $(a,b)$ ...
Aryaan's user avatar
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Seeking feedback on my $\epsilon$-$\delta$ proof of $\lim_{x \to a} x^2 = a^2$.

I'm seeking feedback on my understanding of the $\epsilon$-$\delta$ limit proof for quadratic functions, specifically for $\lim_{x \to a} x^2 = a^2$. After studying multiple proofs, I've noticed that ...
ten_to_tenth's user avatar
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how can I continue from here proving there is no limit to the right of $x_0=0$

when composing g on f at x=0 and checking if there exists A limit to the right of x, I get that there is no limit this is because if I wrote the composition correctly as I take numbers larger than ...
dareen's user avatar
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Adjacent vs. All-Possible Distances in the Definition of a Cauchy Sequence

I'm currently learning about Cauchy sequences, and I'm trying to build my intuition regarding its definition. My question is on how we can motivate why we consider the distances between all points and ...
mouldyfart's user avatar
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1 answer
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Can I plug in numbers for x when proving limits?

At what point can I plug in values for x when proving limits? I know I can't have delta be a function of X. I also know delta should be epsilon divided by 9, but am I allowed to just start plugging in ...
Timothy Pulliam's user avatar
3 votes
1 answer
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Using extra inequalities to prove $\epsilon$-$\delta$ continuity questions.

Context: high school student that read some math books is now reading a calculus book (I'll call it G for the sake of repetition). Still didn't make it to the limit definition. So, there is a ...
Batata's user avatar
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1 answer
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How can any supremum of a proper subset of the real numbers be outside the set, following the logic below?

Let E ⊂ R, E is bounded above, and α = sup E. Then ∀ε > 0, there exists some t ∈ E such that α − ε < t ⇒ α − t < ε Since α > t, α − t = |α − t|. It follows that |α − t| < ε which ...
ranel's user avatar
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0 answers
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In what ways does the properties of infinity alter or invalidate this epsilon-delta proof?

Disclaimer: I am following a math textbook which has not really covered the properties of infinities yet. Only introduced the concept in the context of limits. I have the following problem in my ...
user3612's user avatar
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What is h supposed to represent in the following in the following method for proving limits?

My textbook presents the following general method for proving limits using the delta-epsilon definition: To prove that: $\lim{x\to a}f(x) = L$ Set $x = a + h$ and find the distance: $|f(x) -L| = |f(...
user3612's user avatar
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1 answer
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Limit of function whose inverse has limit approaching infinity at finite point

Consider a continuous strictly monotone function $f:(-1,1)\to\mathbb{R}$, where $$\lim_{x\to1}f(x)=\infty, \lim_{x\to-1}f(x)=-\infty$$ I have proved that this function is bijective. I want to prove ...
Asi Cruz's user avatar
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3 answers
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prove that the function $f(x) = x + \frac{1}{{x^2}}$ is continuous on the interval $(2, \infty)$ using epsilon delta definitions

Let $f(x) = x + \frac{1}{{x^2}}$ and I want to prove that $f$ is continuous at a point $c$ in the interval $(2, \infty).$ I started by defining $\delta$ as $\min(1, \text{ })$ and then I tried to ...
Nicholas Gray's user avatar
2 votes
1 answer
37 views

Prove that if $f$ is continuous, and has two asymptotes, then it is uniformly continuous (Argument check)

The exercise is the following: Let $f:\mathbb{R} \rightarrow \mathbb{R} $ continuous such that $\lim_{x\rightarrow+\infty} f(x) = \ell_1$ and $\lim_{x\rightarrow-\infty} f(x) = \ell_2$ for some ...
Fausto Martinez's user avatar
2 votes
1 answer
68 views

Choosing the infimum in an Epsilon-Delta definition

I was asked to show that the limit of $z^2 = -1$ as $z$ approaches $I$ and my working out is shown below. I am stuck on a few things. Question 1: Why do we choose $\delta$ to be the infimum? Based on ...
Oofy2000's user avatar
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1 answer
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Mistake in solution to show that $\frac{x^ay}{x^2+y^2}$ is continuous at $(0,0)$ if $a>1$.

I was working through some practice problems, and I'm not sure where I'm wrong in the following proof. The function $g:\mathbb{R}^2 \rightarrow \mathbb{R}$ is defined by \begin{cases} \frac{x^ay}...
rudinable's user avatar
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0 answers
52 views

Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$

Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$ See my working below, hopefully you find it legible. Proving it using epsilon-delta, turned it into a proof by contradiction. I assume the limit converges ...
anonymous__'s user avatar
2 votes
4 answers
105 views

How to prove the sequence of function $\{f_n\}$ is Cauchy but not convergent? (Details Below)

Problem Consider the space $C[-1,1]$, together with the norm defined by $\|f\|_1 = \int_{-1}^1|f|d\lambda$ (where $\lambda$ is the Lebesgue measure). For each $n$ define a function $f_n:[-1,1]\to\...
Beerus's user avatar
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3 votes
4 answers
344 views

Proving that a sequence converges, with the epsilon definition

I want to prove that the following sequence $x_n = \frac{3+\sqrt{n}}{2n-\sqrt{n}}$ converges and has a limit. $$\lim_{n \to \infty} \frac{3+\sqrt{n}}{2n-\sqrt{n}} \Rightarrow \lim_{n \to \infty} \...
Noah J. W. 's user avatar
-1 votes
1 answer
65 views

Convergence of the sequence $\frac{3 + \sqrt{n}}{2n - \sqrt{n}}$ with epsilon delta [closed]

How do I prove that this sequence converges: $$ x_n = \frac{3 + \sqrt{n}}{2n - \sqrt{n}} $$ I need to use epsilon delta definition of convergence. I have tried making an upper bound numerator, but it ...
jgt's user avatar
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1 vote
1 answer
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Argumentation of Spivak's choice of epsilon in integrability proof

I have a question about the choice of the bound for uniform continuity (why is it $\frac{\epsilon}{2(b-a)}$) in Spivak's proof of integrability of continuous functions on $[a, b]$. Here is his proof: ...
nz_'s user avatar
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1 vote
0 answers
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Delta-Epsilon proof of $\lim_{x \to 0} \frac{\sin x}{x + 1}$

I've been trying to solve this problem with the delta epsilon proof, but I was unable to do so. I did the following: First, I defined the bound for $\sin x$: $|\sin x| \le 1$ Then, for $\frac{1}{|1+x|}...
Fcatalan's user avatar
-1 votes
2 answers
55 views

Regarding the ϵ-δ definition of limits [duplicate]

The ϵ-δ definition of limits states: Let $ƒ(x)$ be defined on an open interval about c, except possibly at $c$ itself. We say that the limit of $ƒ(x)$ as $x$ approaches c is the number $L,$ and write ...
sab's user avatar
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0 answers
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Uniform convergence of sequence of continuous functions implies uniform continuity?

I recently encountered a question where I'm not sure if I answered it correctly, given that I didn't use one of the assumptions. So here's the question: Prove that if a sequence of continuous ...
Darrell Tan's user avatar
0 votes
1 answer
56 views

Delta Epsilon Question. [duplicate]

My question builds on the well answered question from $\min$ in epsilon-delta and https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/01%3A_Limits/1.02%3A_Epsilon-...
user1115542's user avatar
-1 votes
1 answer
85 views

How do I prove that $a_n \to 0, b_n \to 0 \to a_n + b_n \rightarrow 0$ without pre-established limit rules? [closed]

I've been at this for several hours now and I'm unsure if I'm overthinking it or missing something obvious For the more general case, $a_n \to a, b_n \to b \implies a_n + b_n \to a + b$. I would ...
Subspice's user avatar
1 vote
0 answers
46 views

Is this epsilon-delta proof for a complex limit correct?

I want to prove that: $$\lim_{z\to0}\sqrt z = 0$$ Where $\sqrt{re^{i\varphi}} := \sqrt re^{i\varphi/2}$ is the principal complex square root. Using the following definition: $$\lim_{z\to z_0}f(z)=\ell\...
Elvis's user avatar
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2 votes
2 answers
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Why is $ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy}$ defined while $\lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$ is not?

I've found an exercise that compares the following limits: $$ \lim_{(x,y) \to (0,0)} \frac{\sin(xy)}{xy} \qquad\text{and}\qquad \lim_{(x,y,z) \to (0,0,0)} \frac{\sin(xyz)}{xyz}$$ The solutions ...
vAlkanol's user avatar
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0 answers
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Followup for proof of scalar multiplication of limits

In a previous question of the $\epsilon$ - $\delta$ proof of the scalar multiplication property of limits, they say that $\delta$ is $\frac{\epsilon}{|c|}$ - but why is $c$ absolute? Sorry, quite ...
Chris Sherlock's user avatar
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1 answer
46 views

Structure required for differentiability in topological context

I am a bit unsure about precisely what structure is required for a function to be differentiable, or even continuous for that matter. Specifically, what structure are we using in the $\epsilon-\delta$ ...
Bedge's user avatar
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1 answer
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Determining Whether This Function is Continuous

Is the function $y = x$ if x is rational, and $y = -x$ if x is not rational, continuous at $x = 0$? Thank you in advance. I believe it is so by definition, but it is such a strange case. For any $\...
Xingjian Wang's user avatar
1 vote
1 answer
60 views

Assistance on an $ε-δ$ proof as $x→∞$

I need to provide an $\epsilon$-$\delta$ proof for the following: $$\lim_{x\to \infty}\frac{x^2-4}{x^2-1}=1.$$ My sketch begins with for every $\epsilon>0$,there exists $M>0$ such that $$\left|\...
Nicholas's user avatar

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