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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What is range of values that the word 'nearby' supposed to represent in this informal definition of continuity.
In my book it gave two informal explanation for the concept of continuity. I had doubt in the second explanation but I cleared it by asking it here. The explanation is ,
Suppose a function f has the ...
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Check Understanding of $\varepsilon$-$\delta$ Limit Proof
Say, we want to prove that $\lim_{x \to a} x^2 = a^2\;$ Assuming $a>0$ here.
Here’s how I would think the $\varepsilon$-$\delta$ proof way. Please give feedback on thinking.
$$\forall \varepsilon&...
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Epsilon-delta definition in proving the continuity of $\frac{1}{2}x^2$
How can I prove that the function
$$ f:\mathbb R\rightarrow \mathbb R$$
$$ x\mapsto\frac{1}{2}x^2$$
is continuous?
I currently have:
$$ \epsilon > 0, \delta > 0$$
$$ |x-a| < \delta \...
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How can I prove $\lim_{x \to a} f(x)=L$ implies $\lim_{x \to 3a}f(x/3)=L$ with the epsilon-delta definition of the limit?
How can I prove $\lim_{x \to a} f(x)=L$ implies $\lim_{x \to 3a}f(x/3)=L$ with the epsilon-delta definition of the limit?
I have tried doing this by writing down separately the epsilon-delta ...
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Using the epsilon-delta definition of continuity to prove a linear function is continuous at c
I'm familiar with using the definition to prove a polynomial is continuous at point c, but I've yet to use it with respect to linear functions.
Example:
Prove $h(x)$ is continuous at 4.
$$h(x)=3x-1$$
...
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Clarification about Proof of the Product of two Continuous Functions being Continuous
Almost all $epsilon$-$delta$ proofs of the theorem below either place extra conditions on $\epsilon$, like on page 77 of Tom M Apostol's "Mathematical Analysis," or are copies of the top ...
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Uniformly continuous function implies almost Lipschitz continous.
I wrote a proof of the following exercise but I am dubious there may be an incorrect step in it. I would appreciate your comments:
If $f:\mathbb{R}^n\to\mathbb{R}^m$ is uniformly continuous on $D\...
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$\varepsilon$-$\delta$ limits involving $\sin x$
Currently I'm studying limits by definition and I've seen lots of cases where at some point of the proof, for some given function that involves something of the form:
$\sin [f(x_1 \dots x_n)]$ where $...
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Is my attempt at a proof valid?
Warning: I have no formal education in these, so sorry if my attempt is horrible.
I've recently learnt the concept behind a limit, i.e. the epsilon-delta definition and it seems really cool. I've ...
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Continuity of a pointwise maximum function of probability distributions
Consider a pair of finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. Let $P_{Y|X}$ be a conditional probability distribution and let $Q_Y$ be a full-rank probability distribution.
I am looking at the ...
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Proof Verification for $e^x$ is not uniformly continuous on R
I would like some feedback on my attempt / whether or not this is a valid approach as it differs from the other solutions I have been able to find. I am aware that similar questions have been asked ...
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proving the integral convention $\int_{a}^{b}f(x)dx = \int_{a}^{c}f(x)dx + \int_{c}^{b}f(x)dx$ using the epsilon-delta definition
let $a,b,$ and $c$ be real numbers with $a<c<b$
and function $f$ is integrable in intervals $[a,c]$ and $[c,b]$.
Show that the function f is integrable in $[a,b]$ and that
$\int_{a}^{b}f(x)dx = \...
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Where in this statement is the reverse triangle rule used?
I've attached part of a proof from my lecture notes (the proof is showing that $\frac{1}{x^2}$ is continuous over $\mathbb{R} \setminus \{0\}$), it makes reference to using the reverse triangle rule ...
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Proving $f:\mathbb{R}\setminus \{0\} \to \mathbb{R}, f(x)=\frac{1}{x^2}$ is continuous
We wish to prove that $f:\mathbb{R}\setminus \{0\} \to \mathbb{R}, f(x)=\frac{1}{x^2}$ is continuous by definition.
Scratch work:
$$|f(x)-f(a)| =|\frac{1}{x^2} -\frac{1}{a^2}|=|\frac{x^2-a^2}{a^2x^2}|=...
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Need help with a proof - real analysis/epsilon delta
What I need to prove is the following:
Given the continuous function $ℎ(𝑥)$ on the closed interval $[𝑎,𝑏]$ and $y=\text{inf}\{x: h(x) \geq r\}$, and given the set is non-empty and $𝑦 \in [𝑎,𝑏)$, ...
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Question about a proof in uniqueness of limit in Abbott's "Understanding Analysis"
In Abbott's textbook "Understanding Analysis", first edition, exercise 2.3.4 (page 49), it is asked: "Show that limits, if they exist, must be unique. In other words, assume $\lim a_n=...
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Epsilon delta definition of limit gives seemingly nonsense result
I want to calculate $$\lim_{x\to 0}\frac{\sin\frac1x}{\sin\frac1x}$$
This obviously equals one, but the epsilon delta definition would then be:
For all $\epsilon$>0, there exists a $\delta$>0, ...
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I feel like epsilon-delta is reversed
The lim is about "when x approachs a, then y approachs L". Then, shouldn't the epsilon and delta be like "For all delta, no matter how small the delta is, you can always find an epsilon ...
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How do I prove that $5x^2-3x+\sin(x) \rightarrow 0$ as $x \rightarrow 0^-$ using the $\epsilon-\delta$ definition of limits?
I tried to solve it using the triangle inequality. I ended up with a $\delta$ that wasn't strictly positive, so the proof doesn't work. Here was the attempt:
Fix $\epsilon>0$. Suppose $1>-x>0$...
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Does the limit $\lim_{x\to 0} \sqrt{x^3 - x^2}$ exist or not?
I am having some arguments with a friend about the following limit:
$$\lim_{x\to 0} \sqrt{x^3 - x^2}$$
FACTS: the domain of the function is $x\in \{0\}\cup [1,\ +\infty)$ and $0$ is an isolated point.
...
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How to do Delta Epsilon proofs on limits that need to use L'Hopitals rule?
Hi I'm an 8th grade self studying math and I was curious how would you prove a limit with the $\delta - \epsilon $ that as far as I know needs to use L'Hopitals rule. The limit that I was trying to ...
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$\varepsilon$-$\delta$ proof of $\lim\limits_{x \to 2} x^2 = 4$
$\lim\limits_{x \to 2} x^2 = 4$
$\lvert x-2\rvert \lt \delta$
$\lvert x-2 \rvert \lvert x+2\rvert \lt \varepsilon$
Assuming $\delta$ $\leqslant$ $1$ we find an upper bound for $\lvert x+2 \rvert$ ...
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How can I prove that a multivariable real vector valued function is continuous iff its component functions are continuous?
Suppose we have a multivariable, real, vector-valued function $f:\mathbb{R}^n \to \mathbb{R}^m$ where $f = (f_1, f_2, ..., f_m)$. How can I prove that $f$ is continuous iff all of its component ...
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How to prove function $d_S(x)$ is uniformly continuous
Suppose $S\subset \mathbb R$ is a arbitrary subset and function $d_S(x):\mathbb R\to \mathbb R$ $$d_S(x)=\inf\{|x-a|:a\in S\}
$$ prove this function is uniformly continuous
Since I know the ...
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Real world applications of formal definition of limit. [closed]
Is there a real world application of epsilon-delta definition of limit? I ask it because I want to get more intuitive view on this topic. I mean there are calculations, related to epsilon and delta. ...
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Joint continuity implies separate continuity in $\mathbb R^2$
I need to prove that for a function $f: \mathbb{R^2} \rightarrow \mathbb{R}$ joint continuity at $(x_0, y_0)$ implies separate continuity. I went for an epsilon-delta style proof but have not got ...
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Proving that the limit of product of functions is equal to product of limits with epsilon delta definition
If $\lim_{x\to a} {f(x)}=L$ and $\lim_{x\to a} {g(x)}=M$
Then I wanted to prove that $\lim_{x\to a} {f(x).g(x)}=L.M$
In a book I saw the proof as follows
They initially proved that |f(x)|< 1+|L| by ...
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When a Pseudo-Cauchy sequence converges?
A Pseudo-Cauchy sequence is: A sequence $(a_n)$ if $\forall$ $\epsilon > 0$, $\exists \ N \in \mathbb N$ such that $|a_{n+1} - a_n| \leq \epsilon\;\; \forall n \ge N$.
I know Pseudo-Cauchy ...
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Disproving the existence of a limit using the epsilon delta definition
I wanted to disprove the existence of limit;
$$\lim_{x\to 0} {1\over x} $$
I proved it in the following way.
Let us suppose to the contrary that the limit exists.
We consider 2 cases.
(i) $L≠0$
Let $ε=...
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Proving $\lim_{(x,y)\to (0,0)} \frac{x^{3}-y^{3}}{x^{2}+y^2}=0$ [duplicate]
Let's say I wish to find the limit of $\frac{x^{3}-y^{3}}{x^{2}+y^{2}}$ as $(x,y)\to (0,0)$. It is very simple to just let $y=mx;m\in\mathbb{R}$, and speculate that since the expression comes out to ...
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Increasing functions have a uniform continuity - like property
I am having a hard time solving this problem:
Let $f: [a, b] \rightarrow \Bbb R$ be a weakly increasing function, then:
$$ \forall \epsilon > 0, \exists \delta > 0 : \forall y,x \in [a, b], y-x ...
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Using the epsilon delta criterion to show discontinuity in a point
I am working on the following exercise:
Use the $\epsilon-\delta$-criterion to show that the following function is discontinuous in $0\\$
$sgn(x) = \begin{cases}
-1&,\text{ for }x<0\\
...
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Using $\epsilon$-$\delta$ definition of limit to prove a limit doesn't exist.
How can I use the $\epsilon$-$\delta$ definition of limit to prove that the following limit doesn't exist?
$$ \lim_{x\to1} \sin(\frac{1}{x-1}) $$
So far, I have tried to write out the definition of ...
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Let $g$ be continuous on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one. Show $F$ is either empty or uncountable.
Intermediate Value Theorem (I'll be using this): Let $f : [a,b] \rightarrow \mathbb{R}$ be continuous. If $L$ is a real number satisfying $f(a) < L < f(b)$ or $f(a) > L > f(b)$, then there ...
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Epsilon delta proof of nested sine function
I know that $\lim_{x->0}f(x)=\pi$ and $\lim_{x->\pi}\sin(x)=0$.
I have to prove that $\lim_{x->\pi}f(\sin(x))=\pi$ using the epsilon delta proof.
My idea was to use the value of $\delta=MAX(\...
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Rational Epsilon-Delta Limit Proof Questions
Good afternoon. I have more questions about the details of epsilon-delta proofs. Below is a simple, rational limit proof example with questions at the end. The scratch work and proof are a bit ...
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How can you prove that $\lim_{x \to \infty} \frac{x^{100}}{1.01^x} = 0$ using definition of limit?
How can you prove that $\lim_{x \to \infty} \frac{x^{100}}{1.01^x} = 0$ Using the definition of a limit?
I have been dabbling with this problem, and I am struggling to find an answer. There seem to be ...
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Uniform convergence and rigor proof of pointwise convergence of $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$ for all $x \in (0,\infty)$.
Let $(f_n)$ be a sequence of functions where $f_n:(0,\infty) \to \Bbb R$ defined by $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$.
Show that $f_n$ does not converge uniformly on $(0,\infty)$.
My ...
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Prove that $\lim\limits_{z\to0}z+\vert{z}\vert^3=0$. Epsilon-Delta proof.
The problem is to prove that: $\lim\limits_{z\to 0}z+\vert{z}\vert^3=0$
According to limit rules, we have that $\left[\lim\limits_{z\to z_0}f(z)+g(z)\right]=\lim\limits_{z\to z_0}f(z)+\lim\limits_{z\...
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Proof using Epsilon Delta definition of proof. [Read Description]
I need to prove that
$$\lim_{(x,y)\to(0,0)} \frac{xy(x^2-y^2)}{x+y} = 0. $$
I checked by approaching origin from all directions by substituting $$ y = mx $$ and doing $$\lim_{(x)\to(0)} \frac{xy(x^2-y^...
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Prove that $\lim_{x \to 0} x \sin\left(\frac1x\right)=0$
Hi everyone im struggling a bit with this question. I kinda get the idea of epsilon-delta proof, and have seen you can choose $\epsilon = \delta$ I don't get why tho. Could someone explain that to me?
...
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Simplifying asymptotics.
Question: If for every $\epsilon>0$ however small such that $\epsilon\leq r\leq 1-\epsilon$, then simplify the asymptotic $$f(r)=\mathcal{O}\left(\frac{r}{1-r^2} \ \log\left( \frac{r}{1-r^2} \...
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Proving Riemann Integrability if only if there exists $L \in \mathbb{R}$ (Bartle's exercise 7.1 no.3)
Question :
Show that $f : [a,b]\to \mathbb R$ is Riemann integrable on $[a,b]$ if and only if there exists $L\in \mathbb R$ such that for every $\epsilon \gt 0$ there exists $\delta_\epsilon \gt 0$ ...
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I don't understand what I'm doing wrong: Prove that $f(x) = x^3$ is continuous at $x = −2$.
Prove that $$f(x) = x^3$$ is continuous at $x = −2$.
For this problem, please prove it by imitating the delta -epsilon approach of C1.8.
This are my steps with my solution
$$|(x+2)(x^2-2x+4)|<ε$$
$...
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In epsilon n proofs, how is the index of the sequence related to the expression n>N
I was attempting to understand an epsilon n proof when this was the stated chain of implications:
$$\forall \epsilon > 0, \exists N_2 > 1 : n + 1 \geq N_2 \implies | c_n - L | < \epsilon$$
...
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Is it crucial for an engineering student to "master" the epsilon-delta definitions of limits? [closed]
I know that this is more a discussion than a math question, but I'd like to know. Computer engineering student here. When studying, I like to go through all of the book's definitions and proofs, and ...
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If $f:\mathbb{R}\to\mathbb{R}$ is continuous at $x=0$ and $f(x+y)=f(x)+f(y)$, then $f$ is continuous at every point in $\mathbb{R}$.
Let $f$ be a function defined on all of $\mathbb{R}$ that satisfies the additive condition $f(x+y)=f(x)+f(y)$ for all $x,y \in \mathbb{R}$. I have already proved that $f(0)=0$ and $f(-x)=-f(x)$ for ...
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What is the reason that epsilon delta definition says the result of multiplying upper bound and $|x-a|$ is less than $\epsilon$? [closed]
In epsilon delta definition when we want to prove a limit like $x^2$ at $x = 2$, we should find the upper bound of $|x+2|$ by assuming a random and small value for delta like $1$. next we say the ...
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Find $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2ln(x)}{(x-1)^2 + y^2}$ if it exists.
Show that $\lim_{(x,y)\to (1,0)} \frac{(x-1)^2log(x)}{(x-1)^2 + y^2}$ exists. Also find the limit.
Method 1
$\frac{(x-1)^2}{(x-1)^2 + y^2}$ is less than $1$. Therefore, $\frac{(x-1)^2log(x)}{(x-1)^2 + ...
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why we use open interval in epsilon delta definition fot domain of function?
In epsilon delta definition of limit we see : "Let f be a function defined on some open interval" .
why we use open interval ?
what happen if we use closed interval in definition ?
thanks a ...
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