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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Solving $\lim_{(x,y)\rightarrow (0,0)} \frac{2xy}{x^2+y^2}$ using $\epsilon,\delta$

So we have the following limit to solve, which I solved using multiple paths method (by taking $y=mx$) and investigating the resulting fucntion, and it turned out that limit doesnt exists.$$\lim_{(x,y)...
Kshitij Kumar's user avatar
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Does not the "inexistence of infinitesimals in the real number system" conflict with the epsilon-delta definition of limit in the set of real numbers?

One of the propositions given in Advanced Calculus of a Single Variable is: If |a - b| < ε for each ε > 0 then a = b. Would not this conflict with the epsilon-delta definition of limit? The ...
All is number's user avatar
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How to imagine ( or visualize ) real analysis?

I have a weakness to understand real analysis course, I am not able to imagine these symbols such as epsilon and delta. also, I need to understands and imagine at least every basics theorems in this ...
Math Girl's user avatar
2 votes
3 answers
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Trouble with delta epsilon limit proof of form $\frac{a}{bx+c}$

I want to prove $\lim_{x \to 2} \frac{3}{x+1} = 1 $ using the $ \delta $-$\epsilon$ definition. How can I select a $\delta $ based on $ \epsilon $ so this function stays bounded within 1?
user124820929's user avatar
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Confused with delta epsilon proof of limit

Say I want to prove $$\lim_{x\to1} \frac{1}{(x+1)} = \frac{1}{2}$$ using the $\delta, \epsilon$ - definition. I want to show for all $\epsilon>0$, there exists a $\delta$ such that if $0<|x-1|&...
user124820929's user avatar
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1 answer
64 views

Question about function limit

I have to formally prove $$\lim_{(x,y)\to (2,1)} x^2y=4$$ I am having a hard time getting from $|x^2y-4|$ to two evaluations $|x-2|< \delta_1$ and $|y-1|< \delta_2$. Does someone have some hints,...
Vida Beach's user avatar
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Does order matter in the epsilon-delta definition? [duplicate]

Are the following statements equivalent? Statement 1 For any $\epsilon>0$, there exists $\delta>0$ such that for all real number $x$, if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<\epsilon$ ...
All is number's user avatar
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Does this limit proof about $\lim_{x \to 1} \dfrac{100}{x} = 100$ makes any sense?

I deduced this proof from the proof of the limit of a reciprocal property (Michael Spivak Calculus 3rd edition, pages 103-104). The exercise goes as follows: Exercise. Using epsilon-delta definition, ...
Alexander Mendoza's user avatar
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Proof of Cesàro summation

This is a proof I came up with while working on the textbook Understanding Analysis: Supposing $x_{n} \rightarrow x$, we have that $$s_n = \frac{1}{n} \sum_{k=1}^{n} x_k \rightarrow x$$ Let $\epsilon \...
Mani's user avatar
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A cool function that connects the derivative to the modulus of continuity

While studying analysis, an interesting question came up: Prove that if a function $f: [a,b]\to \mathbb{R} $ Is continuous, where $[a,b]\subseteq \mathbb{R}$, then $f$ is uniformly continuous. To ...
Carlyle's user avatar
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Prove the limit of the below function with epsilon-delta definition [closed]

How do I prove the limit of following function using the epsilon-delta definition 1
kavin's user avatar
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can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?

I’m taking a class in Complex Analysis and trying to prove the same result as found in this question. In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those ...
ekorel's user avatar
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Prove $\lim_{x\to 1} \frac{x-1}{2x^2+x-3} = \frac{1}{5}$ using the $(\epsilon, \delta)$-definition.

I want to see if my proof is correct and if my choice of $\delta$ makes sense. Prove that $\lim_{x\to 1} \frac{x-1}{2x^2+x-3} = \frac{1}{5}$. Let $\delta = \min(1, \frac{15}{2} \epsilon)$ and suppose $...
frozencoy's user avatar
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The "same old" dy/dx question? Separation of differentials?

I completely understand this question has been addressed one too many times. But I still simply cannot wrap my head around the concept of dy/dx. Simply put, when can we treat dy/dx as a ratio and when ...
S_M's user avatar
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Multivariable delta-epsilon limit proof

Prove using the delta epsilon definition that $$ L = \lim_{(x,y)\rightarrow (0,0)}\frac{x^5e^{xy}}{x^2+e^y}=0 $$ What I have done so far: $$\left | \frac{x^5e^{xy}}{x^2+e^y}-0 \right |< \epsilon \;...
Billy Walsh's user avatar
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1 answer
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Epsilon Delta Proof of the Derivative of X Cubed

Right, so this was a question I randomly pondered up while I was watching through a BlackPenRedPen video, in which he shows off a question similar to the one I'm about to ask, except with f(x) = $x^2$ ...
LogicBeDamned's user avatar
3 votes
2 answers
132 views

Proving the limit $\lim_{x \to -2} \frac{\sqrt[3]{x}+\sqrt[3]{2}}{x+2}=\frac{\sqrt[3]{2}}{6}$ using the epsilon-delta definition.

Prove $\lim_{x \to -2} \frac{\sqrt[3]{x}+\sqrt[3]{2}}{x+2}=\frac{\sqrt[3]{2}}{6}$. Solution Attempt: $\left| f(x) - L \right| < \epsilon$ $\left|f(x)-L \right|=\left| \frac{\sqrt[3]{x}+\sqrt[3]{2}}{...
남성 심장's user avatar
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When picking a delta to show a function is continuous, does that delta need to rely on epsilon?

Generally, when we want to show continuity, we are given a small $\varepsilon$ and we pick a $\delta$ according to this $\varepsilon.$ Heuristically, we can consider this $\delta$ to be a function of $...
ikey's user avatar
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Where was the formal definition of a limit first published?

I've read that Bolzano was one of the first to formalize a limit with the epsilon-delta definition, but I can't find a text of his that shows his definition. Does anyone know what publication the ...
Jenna Sickau's user avatar
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In epsilon-delta proofs, is delta found be equal to a value or less than or equal to a value?

In epsilon-delta proofs, when specifying delta, is it correct to specify the delta is equal to a value or delta is less than or equal to a value? Which statement if more correct and why? Given epsilon ...
bamajon1974's user avatar
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Supremum of all deltas for a given epsilon

A function $f:\mathbb{R} \to\mathbb{R}$ is continuous at a point $c$ if, for every $\varepsilon >0$ there exists a $\delta > 0$ such that $|x-c| < \delta$ implies $|f(x)-f(c)|<\varepsilon$....
M. Sperling's user avatar
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Prove that a family of functions bounded by a $L^1$ function is uniformly integrable: $\varepsilon$-$\delta$ proof

Let $\mu$ be a finite measure and let $(f_i)_{i\in I}$ be a sequence of functions such that $$ f_i\le (k g)^{1/2} \quad \forall i\in I, $$ where $k>0$ is a constant and $g\in L^1(\mu)$. I am trying ...
Physics user's user avatar
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Proving every continuous function on closed interval is uniformly continuous by smallest delta

When proving the statement above, I completely understand the textbook proof with convergent subsequences and contradiction, but I wonder: when function is continuous by classic epsilon-delta ...
mathion's user avatar
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What does $V_{\delta_x} (x) $ mean?

I was solving exercises on real analysis by Introduction to Real Analysis by Robert Bartle and Donald Sherber forth edition exercise 5.3 page 141 and I am confused what does $V_{\delta_x}(x)$ in Let ...
pie's user avatar
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Assumptions when proving limit using epsilon delta definition

Let’s say I have a limit for example $\lim_{x\to 4}x^2 + x -11= 9$ and I want to prove this limit using the epsilon-delta definition. So then $|x-4| < \delta$ and $|x^2+x-20| < \epsilon$ Now $|x+...
Dan Lupu's user avatar
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Limit of $x[3−\cos(x^2)]$ using the epsilon-delta definition

This is a duplicate of this post here, but I'm new to Stack Exchange so I'm not able to comment on that post yet (and sorry in advance if this question isn't appropriate to the platform). I was ...
cb2's user avatar
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2 answers
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Proving $\lim_{x\to 1} \frac{x^2}{x^2 + 1} = \frac12$ using $\delta$-$\varepsilon$ method [duplicate]

Question: Prove $\lim\limits_{x\to 1} \frac{x^2}{x^2 + 1} = \frac12$ using $\delta$-$\varepsilon$ method. Source: I recently came across this post and I am a beginner in understanding $\delta$-$\...
Utkarsh's user avatar
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Continuity at a point doesn’t imply continuity at some small interval around it

I wanted to prove that point continuity does not imply local continuity. I have already seen examples like the function $f(x) = x^{2}$ if $x$ is rational and 0 if irrational, which are nowhere ...
Alejandro's user avatar
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1 answer
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proving $ \lim_{x\to \infty}$ ${x+3\over x-2}$=1 using delta epsilon method

I need to prove $ \lim_{x\to \infty}$ ${x+3\over x-2}$=1 using delta-epsilon method. Here's the part I tried but I'm stuck in the middle in choosing M with $\varepsilon$. Let $\varepsilon$ >0 be ...
dkSs's user avatar
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3 answers
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Why is the limit of a function defined at only a cluster point of the domain?

My textbook gives the following definition of a limit: Let $A\subseteq\mathbb R$, and let $c$ be a cluster point of $A$. For a function $f:A\to\mathbb R$, a real number $L$ is said to be the limit of ...
user1206158's user avatar
1 vote
1 answer
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Nature of the set of deltas in limit definition

I was proving the fact that if f is differentiable at a, then (cf)’(a)=cf(a)’ After showing it, I started to play with some ideas and wondered from intuition how to see that depending on the value of ...
Alejandro's user avatar
2 votes
3 answers
76 views

Prove that $\lim_{n\to\infty}a_n=1/2$ by definition

Suppose that $$ a_n=\frac{n^2-2n+1}{2n^2+4n-1}. $$ I am trying to show that for each positive number $\epsilon$, there is a number $N$ such that $|a_n-L|<\epsilon$ whenever $n>N$. Attempt ...
Bell's user avatar
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Algebraic rules of absolute value

In evaluating elementary $\epsilon$, $\delta$ proofs of limits, one often sees the following sort of move: $$ \left|2x - 8\right| = \left|2(x-4)\right| = 2\left|x - 4\right| \dots$$ (See e.g. here (14:...
RTF's user avatar
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Continuity of $f(n):=n,6n\in\mathbb{Z}$

Suppose $f:\frac{1}{6}\mathbb{Z}\to\frac{1}{6}\mathbb{Z}$ is a function defined by $f(n):=n,6n\in\mathbb{Z}$. Is $f$ continuous at $x=0$? A graph of the function somewhat points towards this answer: ...
Ark1409's user avatar
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2 answers
157 views

Proof of limit by epsilon-delta method: $\displaystyle\lim_{x \to 9} \sqrt{x-5} = 2$

The question is: Prove the below limit statement: $\displaystyle\lim_{x \to 9} \sqrt{x-5} = 2$ From my understanding of the textbook (Thomas' Calculus), the proof is done in 3 steps: Write both the ...
Advaith's user avatar
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-1 votes
3 answers
273 views

Epsilon delta is super redundant, isn't it? [closed]

I have a very different question. I was looking into $\epsilon-\delta$ definition of limit. I actually understand the idea, but what I'm wondering is why it was necessary to come up with such an idea ...
Chemistry's user avatar
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1 answer
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Solving quadratic inequalities which cannot be factorized

This question is a step involved in solving the below question: Find an open interval about c on which the inequality $|f(x)-L|<\epsilon$ holds. $f(x) = 4+x-3x^2$, $L=2$, $c=1$, $\epsilon=0.01$ I ...
Advaith's user avatar
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1 answer
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Finding an upper bound of $\delta$ in the $\delta-\epsilon$ proof when the function is differentiable

I'm reviewing my real analysis and I came across this statement: Let $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function, and let $c\in \mathbb{R}$. Then if $x\in B(c,\epsilon/f'(c))$, we have $...
Tianyi Wang's user avatar
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1 answer
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Using epsilon-balls on metric spaces

Need help with a metric space epsilon ball question.. If $\ell_1$ is the set of all sequences of real numbers $(x_n)_{n=1}^\infty$ such that $\sum_{i=1}^{\infty} |x_i| < \infty$ with metric $$d_1((...
Craig Lutic's user avatar
1 vote
0 answers
58 views

Is a function discontinous at points not in its domain? [duplicate]

Famously, $\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x} = 1.$ However, whether or not it is considered continuous at $x = 0$ seems unclear to me. Another post has pointed out that Rubin's Principles of ...
Henry Zhang's user avatar
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1 answer
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The quotient of two series

I have trouble solving the following problem: The positive series $\sum_{n=0}^\infty a_n$ converges, and $\lim_{n\to \infty}\frac{b_n}{a_n}=1$. Prove that the series $\sum_{n=0}^\infty b_n$ converges ...
QIRUN CONG's user avatar
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Uniform continuity subtlety

I have studied the concept of uniform continuity of a function , and I have been doubting on the following: If the function is continuous on the interval [a,b], we know that for every point y in it, ...
Alejandro's user avatar
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1 answer
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Proving the continuity of upper and lower integral.

In this problem, we will prove that if $f: [a,b] \rightarrow \mathbb{R}$ is bounded, then the functions $G(x) = \overline{\int_{a}^x}f(t) dt$ and $H(x) = \underline{\int_{a}^x}f(t) dt$ are continuous....
Aristarchus_'s user avatar
15 votes
13 answers
3k views

Does the epsilon-delta definition of limits truly capture our intuitive understanding of limits?

I've been delving into the concept of limits and the Epsilon-Delta definition. The most basic definition, as I understand it, states that for every real number $\epsilon \gt 0$, there exists a real ...
thomas graceman's user avatar
-1 votes
1 answer
220 views

$\epsilon$-$\delta$ proof that $\lim_{x \to 0}\frac{e^x-1}{x} = 1$ without using $\frac{d}{dx}e^x = e^x$

Here is my attempt at finding a proof that $\lim_{x \to 0}\frac{e^x-1}{x} = 1$. We need to show that $\forall \varepsilon > 0 \, \exists \delta > 0$ such that $ 0 < |x| < \delta \implies |...
Shuhul Mujoo's user avatar
2 votes
1 answer
95 views

Example of a limit where the Epsilon delta definition wins

In my high school Limit of a function was introduced as below. We say $\lim_{x\to a^-} f(x)$ is the expected value of $f$ at $x = a$ given the values of $f$ near $x$ to the left of $a$. This value is ...
user534666's user avatar
1 vote
1 answer
25 views

Finite decimal continuity

I am learning multi-var calculus using the textbook "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach". Here is a definition that's puzzling me. Let $\mathbb{D}$ ...
Mani's user avatar
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3 votes
3 answers
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can we say a function $f:[a,b]\to \Bbb{R}$ is continuous at the end points?

$f:[a,b]\to \Bbb{R}$ is said be continuous at a point $m$ that means whenever a sequence $x_n \in[a,b]$ converge to ${m}$ the image ${f(x_n)} $ converge to ${f(m)}$ and by applying this definition of ...
pie's user avatar
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1 vote
1 answer
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Does the existence of a limit imply that a function has a maximum/minimum in that neighbourhood?

My question arises from when I'm trying to prove that if $\lim_{x\rightarrow a}f(x)=m$ ($m≠0$) then $\lim_{x\rightarrow a}\frac{1}{f(x)}=\frac{1}{m}$ using epsilon delta. Since $\lim_{x\rightarrow a}f(...
Pen and Paper's user avatar
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2 answers
46 views

Understanding $\epsilon-\delta$ definition of continuity [duplicate]

$\textbf{Textbook defenition:}$ Let $A$ be a subset of $\mathbb{R}$. Let $f: A \rightarrow \mathbb{R}$ and let $c \in A$. We say that $f$ is continuous at $c$ if, given any number $\epsilon>0$ ...
Ellie_Wong's user avatar

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