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.$$\lim_{(x,y)...
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Does not the "inexistence of infinitesimals in the real number system" conflict with the epsilon-delta definition of limit in the set of real numbers?
One of the propositions given in Advanced Calculus of a Single Variable is:
If |a - b| < ε for each ε > 0 then a = b.
Would not this conflict with the epsilon-delta definition of limit?
The ...
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How to imagine ( or visualize ) real analysis?
I have a weakness to understand real analysis course, I am not able to imagine these symbols such as epsilon and delta. also, I need to understands and imagine at least every basics theorems in this ...
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Trouble with delta epsilon limit proof of form $\frac{a}{bx+c}$
I want to prove
$\lim_{x \to 2} \frac{3}{x+1} = 1 $ using the $ \delta $-$\epsilon$ definition. How can I select a $\delta $ based on $ \epsilon $ so this function stays bounded within 1?
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Confused with delta epsilon proof of limit
Say I want to prove $$\lim_{x\to1} \frac{1}{(x+1)} = \frac{1}{2}$$
using the $\delta, \epsilon$ - definition. I want to show for all $\epsilon>0$, there exists a $\delta$ such that if $0<|x-1|&...
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Question about function limit
I have to formally prove $$\lim_{(x,y)\to (2,1)} x^2y=4$$ I am having a hard time getting from $|x^2y-4|$ to two evaluations $|x-2|< \delta_1$ and $|y-1|< \delta_2$. Does someone have some hints,...
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Does order matter in the epsilon-delta definition? [duplicate]
Are the following statements equivalent?
Statement 1
For any $\epsilon>0$, there exists $\delta>0$ such that for all real number $x$, if $|x-x_0|<\delta$, then $|f(x)-f(x_0)|<\epsilon$
...
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Does this limit proof about $\lim_{x \to 1} \dfrac{100}{x} = 100$ makes any sense?
I deduced this proof from the proof of the limit of a reciprocal property (Michael Spivak Calculus 3rd edition, pages 103-104). The exercise goes as follows:
Exercise. Using epsilon-delta definition, ...
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Proof of Cesàro summation
This is a proof I came up with while working on the textbook Understanding Analysis:
Supposing $x_{n} \rightarrow x$, we have that
$$s_n = \frac{1}{n} \sum_{k=1}^{n} x_k \rightarrow x$$
Let $\epsilon \...
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A cool function that connects the derivative to the modulus of continuity
While studying analysis, an interesting question came up:
Prove that if a function
$f: [a,b]\to \mathbb{R} $
Is continuous, where $[a,b]\subseteq \mathbb{R}$, then $f$ is uniformly continuous.
To ...
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Prove the limit of the below function with epsilon-delta definition [closed]
How do I prove the limit of following function using the epsilon-delta definition 1
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can epsilon in a epsilon-delta proof rely on the independent variable if it cancels out at the end?
I’m taking a class in Complex Analysis and trying to prove the same result as found in this question.
In the accepted answer, a delta is chosen such that it’s the minimum of two values - one of those ...
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Prove $\lim_{x\to 1} \frac{x-1}{2x^2+x-3} = \frac{1}{5}$ using the $(\epsilon, \delta)$-definition.
I want to see if my proof is correct and if my choice of $\delta$ makes sense.
Prove that $\lim_{x\to 1} \frac{x-1}{2x^2+x-3} = \frac{1}{5}$. Let $\delta = \min(1, \frac{15}{2} \epsilon)$ and suppose $...
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The "same old" dy/dx question? Separation of differentials?
I completely understand this question has been addressed one too many times. But I still simply cannot wrap my head around the concept of dy/dx.
Simply put, when can we treat dy/dx as a ratio and when ...
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Multivariable delta-epsilon limit proof
Prove using the delta epsilon definition that
$$
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 \;...
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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) = $x^2$ ...
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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}}{...
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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 $...
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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 ...
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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 ...
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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 $|f(x)-f(c)|<\varepsilon$....
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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 ...
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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 ...
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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 ...
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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+...
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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 ...
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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 $\delta$-$\...
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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 ...
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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 ...
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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 ...
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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 ...
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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
...
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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:...
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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:
...
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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 ...
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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 ...
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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 ...
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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 $...
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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((...
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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 ...
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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 ...
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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, ...
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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....
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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 ...
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$\epsilon$-$\delta$ 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 |...
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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 ...
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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 $\mathbb{D}$ ...
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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 ...
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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(...
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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 $\epsilon>0$ ...
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