Questions tagged [epsilon-delta]

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

Filter by
Sorted by
Tagged with
2 votes
1 answer
36 views

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 ...
user avatar
0 votes
1 answer
80 views

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&...
user avatar
0 votes
2 answers
53 views

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 \...
user avatar
0 votes
2 answers
39 views

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 ...
user avatar
1 vote
1 answer
25 views

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. $$h(x)=3x-1$$ ...
user avatar
  • 65
0 votes
0 answers
16 views

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 ...
user avatar
-1 votes
0 answers
18 views

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\...
user avatar
0 votes
1 answer
53 views

$\varepsilon$-$\delta$ 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 $...
user avatar
1 vote
2 answers
76 views

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 ...
user avatar
  • 57
0 votes
0 answers
11 views

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 ...
user avatar
  • 1,528
0 votes
2 answers
46 views

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 ...
user avatar
  • 991
0 votes
1 answer
38 views

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 = \...
user avatar
  • 113
2 votes
2 answers
37 views

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 ...
user avatar
0 votes
1 answer
59 views

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}|=...
user avatar
0 votes
1 answer
31 views

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 $𝑦 \in [𝑎,𝑏)$, ...
user avatar
  • 105
2 votes
1 answer
50 views

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=...
user avatar
  • 379
0 votes
1 answer
64 views

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, ...
user avatar
11 votes
2 answers
695 views

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 ...
user avatar
0 votes
1 answer
40 views

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 $1>-x>0$...
user avatar
5 votes
2 answers
163 views

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. ...
user avatar
1 vote
1 answer
57 views

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 ...
user avatar
  • 23
0 votes
1 answer
46 views

$\varepsilon$-$\delta$ 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 $\lvert x+2 \rvert$ ...
user avatar
0 votes
1 answer
54 views

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 ...
user avatar
  • 131
1 vote
2 answers
46 views

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 ...
user avatar
  • 754
-6 votes
1 answer
59 views

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. ...
user avatar
1 vote
0 answers
42 views

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 ...
user avatar
  • 11
0 votes
1 answer
31 views

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 ...
user avatar
  • 125
1 vote
0 answers
40 views

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 ...
user avatar
2 votes
1 answer
68 views

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=...
user avatar
  • 125
0 votes
0 answers
48 views

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 ...
user avatar
  • 6,273
0 votes
1 answer
48 views

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 ...
user avatar
0 votes
1 answer
69 views

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\\ ...
user avatar
0 votes
1 answer
81 views

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 ...
user avatar
  • 359
1 vote
1 answer
102 views

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 ...
user avatar
  • 85
0 votes
2 answers
45 views

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(\...
user avatar
2 votes
1 answer
74 views

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 ...
user avatar
1 vote
2 answers
99 views

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 ...
user avatar
1 vote
2 answers
64 views

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 ...
user avatar
  • 1,951
0 votes
1 answer
74 views

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\...
user avatar
0 votes
1 answer
41 views

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^...
user avatar
  • 1
1 vote
3 answers
74 views

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? ...
user avatar
  • 31
1 vote
0 answers
100 views

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} \...
user avatar
0 votes
0 answers
41 views

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 $\delta_\epsilon \gt 0$ ...
user avatar
  • 55
0 votes
2 answers
89 views

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 $$|(x+2)(x^2-2x+4)|<ε$$ $...
user avatar
  • 19
0 votes
2 answers
38 views

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: $$\forall \epsilon > 0, \exists N_2 > 1 : n + 1 \geq N_2 \implies | c_n - L | < \epsilon$$ ...
user avatar
  • 105
2 votes
3 answers
465 views

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 ...
user avatar
2 votes
1 answer
81 views

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 ...
user avatar
  • 85
0 votes
1 answer
51 views

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 ...
user avatar
  • 15
0 votes
0 answers
34 views

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 + ...
user avatar
1 vote
2 answers
94 views

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 ...
user avatar
  • 15

1
2 3 4 5
52