Questions tagged [epsilon-delta]
For questions regarding $\varepsilon$-$\delta$ definitions of limits and continuity.
0
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1answer
19 views
A formal epsilon-delta proof for the Continuity Law for Composition
The continuity law for composition states, informally, that:
$$\text{IF } f \text{ is continuous at } g \ \text{ AND }\ g \text{ is continuous at } f(a) \text{ THEN } g(f(a)) \text{ is continuous at }...
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votes
2answers
53 views
Proving $\lim_{x\to9}\sqrt{x-5}=2$
Im stuck on proving $\displaystyle \lim_{x\to9}\sqrt{x-5}=2.$
I start with let $e>0$ be given we need to find a $d>0$ such that whenever
$0<|x-9|<d$ we have $|f(x)-2|<e$
My method is ...
3
votes
3answers
86 views
Why can't we assume $0 < |x - a| \leq \delta$ (equality) in an epsilon delta proof?
Suppose we want to prove that
$$ \lim_{x \to a} f(x) = L,$$
that is
$$ \forall \epsilon > 0, \exists \delta > 0, \forall x \in \mathbb{R}, 0 < |x - a | < \delta \Longrightarrow |f(x) - ...
2
votes
2answers
43 views
is my epsilon-delta proof correct?
I am a complete beginner with limits and I self study, so I don't have anyone to confirm my answers. I had a simple limit to prove with the precise definition, its a linear equation and I did lots of ...
2
votes
3answers
76 views
Proving $\lim_{x\to \infty} xe^{-x^2} = 0$ using definition
I am trying to prove that $\displaystyle\lim_{x\rightarrow \infty} xe^{-x^2} = 0$ using the formal definition of limits as $x$ approaches infinity. The definition I am using is as follows:
$$\lim_{x\...
-1
votes
3answers
40 views
Prove $\lim_{(x,y)\to(0,0)} \frac{xy^2}{x^2+y}$ [on hold]
Prove $\lim_{(x,y)\to(0,0)} \frac{xy^2}{x^2+y}=0$ using epsilon-delta proof.
I tried to solve this problem with epsilon-delta method, but couldn't handle it.
anyone help me?
2
votes
1answer
65 views
Show that $X_n \to 0$ in probability under given condition.
Let $k > 0$. Suppose that
$$\forall \epsilon > 0: \exists N: \forall n \geq N: P(|X_n| \geq \epsilon) \leq \epsilon k$$
Show that $X_n \xrightarrow{P}{ 0}$.
Attempt:
We have to show:
$$\...
0
votes
2answers
35 views
Is there an error in this epsilon-delta proof of the limit law for the sums of functions?
One of my Calculus profs recorded a video (screenshot below) to prove the limit law for the sums of functions:
$$\lim_{x\to a} f(x) = L \ \wedge \ \lim_{x\to a} g(x) = M \Longrightarrow \lim_{x\to ...
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votes
2answers
35 views
How to prove the continuity? [duplicate]
Prove that the function :
$f(x) = x$ when $x$ is rational and
$f(x) = 1 - x$ when $x$ is irrational
is continuous at $x = \frac{1}{2}$
To prove the continuity
it is necessary to prove that for every $...
5
votes
1answer
328 views
What does it mean for something to be strictly less than $\epsilon$ for an arbitrary $\epsilon$?
Perhaps a trivial question, but something I never completely understood. If we have shown that $a-b < \epsilon$ for all $\epsilon > 0$, then does that imply that $a-b \le 0$?
I"m interested in ...
1
vote
2answers
33 views
Showing that equilibrium is unstable
I just solved an ODE $x'=A(t)x$ which has $x(t)=e^{t/2}(-\cos t, \sin t)^T$ as a solution.
Now I want to show that the equilibrium $\bar x=0$ is unstable.
So according to my book I need to show that ...
0
votes
0answers
18 views
Show that (implicit) differentiable map with invertible derivative has open image
I'm doing a homework exercise with several steps.
Let $\Phi: \mathbb{R}^n\supset U\to \mathbb{R}^n$ be a differentiable map with invertible derivative $D\Phi(x)$ for any $x\in U$. Let $a\in U$ and $b:...
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0answers
21 views
Proving a sequence does not converge by the definition.
Question:Prove that this sequence does not converge for any $x \in \Bbb R.$
$x_n=(-1)^n(1-\frac{1}{n})$
Definition (Negation):$ \exists \varepsilon \forall N \in \Bbb N \exists n \in \Bbb N, n>N:...
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0answers
23 views
Proving a piece-wise function does not converge
Question: Does the function $f(x)=\{1$ if $x\in \Bbb Z $, $0$ if $x \notin \Bbb Z$, tend to zero as $x$ tends to infinity?
Not sure how to do the piece-wise function, feel free to edit ...
1
vote
1answer
38 views
Formal proof that $\lim_{(x,y) \to (0,1)} \frac{x}{\sqrt{y}} =0$
for midterm preparation I am trying to prove that:
$$\lim_{(x,y) \to (0,1)} \frac{x}{\sqrt{y}} =0$$
Using the formal $\epsilon - \delta$ definition. I am having trouble with the estimates, Suppose $\...
0
votes
2answers
28 views
Prove that if a sequence $x_n$ tends to infinity, then $\frac{1}{x_n}$ converges to zero.
Question: $(x_n)_{n=1}^\infty$ is a sequence with $x_n\neq0 $ for all $n$, also let $x_n$ tend to infinity. Let $(y_n)_{n=1}^\infty$ be defined by $y_n=\frac{1}{x_n}$, show that this converges to zero....
2
votes
0answers
21 views
Sequence tending to infinity (checking epsilon proof)
Question: Prove that the sequence $(x_n)_{n=1}^\infty$ defined by $x_n=\sqrt[3]{n}+1$ tends to infinity.
Definition: $\forall K \in \mathbb{R} \exists N\in \mathbb{N} \forall n \in \mathbb{N} ,n>...
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votes
2answers
45 views
Epsilon delta proof for this function
I am having a lot of trouble finding an epsilon delta proof for the following: $$\lim_{t\rightarrow0} \hspace{5px} \dfrac{\sin(t^2)}{2t^2} = \dfrac{1}{2}$$
I have tried a lot of things and I can show ...
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votes
1answer
48 views
Prove using epsilon-delta definition that $\lim_{x \to 0} \frac{1}{x^2+1}=1$ [closed]
I need to prove using the epsilon-delta definition that: $$\lim_{x \to 0} \frac{1}{x^2+1}=1$$
I understand the definition, and what it implies, but I’m having some troubles at the moment of writing ...
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votes
5answers
81 views
Proving that $\lim_{x\rightarrow -\infty}e^x=0$ with epsilon delta definition
I know that we need to find a way of relating $|x+\infty|<\delta$ and $|e^x-0|<\epsilon$. So, $|e^x-0|<\epsilon \iff e^x<e^{\log(\epsilon)}\iff x<\log(\epsilon)$. Then I imply that $\...
0
votes
2answers
29 views
Is this a valid proof that $(a_n) \rightarrow l$ implies $(\sqrt{a_n}) \rightarrow \sqrt{l}$?
Let $(a_n)$ be a sequence of non-negative real numbers converging to $l>0$. Show that $\lim_{n\rightarrow\infty}\sqrt{a_n} = \sqrt{l}$.
Fix $\epsilon>0$.
$\exists N\in \mathbb{N}$ s.t. $n\geq ...
2
votes
1answer
32 views
What does the symbol $\delta$ mean on this page?
On the Wikipedia page on Arithmetic Functions, the section Relations Among The Functions makes frequent references to a variable $\delta$ (or is it a function? Some other kind of value?).
It's ...
1
vote
1answer
46 views
Proving that $\lim_{(x,y)\to(0,2)} |y|^x(x+1)^y = 1$ using the limit definition
I started with this strategy:
$$||y|^x(x+1)^y - 1| = \bigg||y|^x \bigg((x+1)^y -\frac{1}{|y|^{x-1}} + \frac{1}{|y|^x} \bigg) + |y| - 2 \bigg| \leq \bigg||y|^x \bigg((x+1)^y -\frac{1}{|y|^{x-1}} + \...
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votes
2answers
32 views
Bounding denominators in $\epsilon-\delta$ limit proofs
I want to prove that
$$\lim_{(x,y)\to(1,0)} \frac{x-y}{x^2+y^2} = 1$$
But I'm out of ideas:
$$\bigg|\frac{x-y}{x^2+y^2} - 1\bigg| = \bigg|\frac{x-y-x^2-y^2}{x^2+y^2} \bigg| \leq \frac{|x||x-1| + ...
1
vote
1answer
27 views
Find Delta, given Epsilon lim x->0 root (x+1) = 1. epsilon = 0.1
Can I solve this using rationalization and then assuming delta <= 1? If so, why is my answer wrong?
we know |√(x+1)-1| < 0.1
rationalization of LHS,
x<0.1*(|√(x+1)+1|)
hence, delta = 0.1*(|√(...
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votes
1answer
29 views
While finding delta algebraically of quadratic functions, can we proceed in this way?
while doing it this way, the answer obtained is wrong. Where have I possibly made an error?
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0answers
45 views
Proving limits using the $\epsilon-\delta$ definition [closed]
Are there any guides on how to prove complicated limits by definition (either online or in a textbook) that provide both exercises and solutions?
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1answer
31 views
Proving that if $\lim_{(x,y)\to(a,b)} f(x,y) = \infty$, then $\lim_{(x,y)\to(a,b)} \frac{ln(f(x,y))}{f(x,y)}=0$
How can I properly prove this using definitions? Does the hypothesis allow me to get rid of the denominator in $|\frac{\ln(f(x,y))}{f(x,y)}|$?
1
vote
1answer
32 views
Proof Checking: Corollary epsilon delta definition of limit
I'm trying to solve this problem:
Suppose that $\lim_{x \rightarrow c} f(x)$ exists. Prove that there exists a constant $M$ and a $\delta > 0$ such that $|f(x)|<M$ for $0 < |x - c| < \...
2
votes
2answers
57 views
Guessing delta value, in epsilon-delta proof
So as I've solved various problems regarding epsilon-delta proof, I have faced several questions where I had to kind of implement this process:
So here's one of them:
$\lim_{x\rightarrow 1} (x^3+x+1)...
1
vote
1answer
33 views
Prove that $f(x, y) = 2x - 3y$ is continuous using epsilon-delta
I'm a bit rusty on epsilon delta proofs so I wanted to double check that my proof is correct:
Let $\epsilon > 0$ and $\delta = \epsilon * \frac{\sqrt{2}}{3}$. Then for $(x,y)$ and $(u,v)$, $\sqrt{(...
2
votes
1answer
37 views
Absolute Value within an Epsilon Delta Problem
I'm having trouble with an epsilon-delta proof. I'm asked to prove the limit as $x$ goes to $-2$ of $f(x)=|x^2-9|/(x^2+3x+1)$ is $-5$, so we have $|(|x^2-9|/(x^2+3x+1))-5|<ϵ$. I've seen that $|x^2-...
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votes
2answers
24 views
Unclear about the epsilon-delta definition of continuous mapping
In short, I don't see how the epsilon-delta definition excludes non-injective mappings. For example, I can imagine a modified sine where a single point is excluded, but for which we can still find the ...
1
vote
1answer
38 views
Epsilon-N proof [closed]
Hello I want to prove that
$$\frac{1+\sin n}{n^2+1}\rightarrow0 \text{ as } n\rightarrow\infty$$
Using the Epsilon-N proof. I'm not sure what to do during the scratchwork. How do I work with the $\...
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votes
0answers
35 views
Prove Series Function is Continuous
How can I prove that series function is continuous with epsilon-delta definition.
$f(x)=\sum_{n=1}^{\infty} \frac{ {\displaystyle \lfloor nx\rfloor } }{n^{2}}$
a) Prove that it is discontinuous in ...
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vote
1answer
27 views
Proving that a function is discontinuous using sequential definition
I am struggling to understand how to prove that a function is discontinuous using the sequential definition. Here is a particular example from my textbook where some clarification might help.
Let f(x)...
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votes
1answer
39 views
Real Analysis: The continuity of $\frac {1}{1-x}$ on the interval $[0,1).$
Need help to prove the continuity of the function $f(x)=\frac {1} {1-x}$ on $[0,1).$ Using the epsilon-delta definition, I came to $\frac {|x-x_0|} {|1-x||1-x_0|}$ and I don't know how to proceed ...
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votes
0answers
24 views
Epsilon delta proof with ln and sin(x)
I tried it without using the fact |sin x| <= |x| for all x, but I know using this, would make it way easier. In fact, I'm not even sure if my proof is correct.
Proof:
Let ε > 0 be given. We want ...
0
votes
1answer
32 views
Intermediate Value Theorem. Why $x=s+𝛿/2$
I've read the Intermediate Value Theorem's proof and I understand everything except one line which i've been speculating how it is derived. I know this may be a silly question but it does confuse me. ...
1
vote
0answers
38 views
Prove this sequence is Non-Cauchy
I'm trying to solve this question but am finding it difficult to prove. I am using the negation of the Cauchy definition for sequences
$\exists 𝝐>𝟎, \forall 𝑵∈\mathbb Z^+, \exists n,m>N, |...
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votes
1answer
16 views
Epsilon Delta proof containing decimal exponents
so I have to prove
lim x->infinity ((x^0.8)/(1+x^0.9)) = 0
I am just introduced to epsilon delta, and have no idea how to do this.
Please help :( Thanks!
1
vote
1answer
55 views
Elementary proof of $\lim\limits_{x\to 0}a^x = 1$ when $a>0$
I have a proof which uses sequences that converge to $0$.
I think there is an easier proof than mine but I couldn't find.
Is there anyone can prove this without using sequences?
Or would you please ...
0
votes
1answer
34 views
Can continuous functions have removable discontinuities?
I'm trying to resolve what seems like an inconsistency between the epsilon-delta definition of continuity and the limit-based definition ($\lim_{x->c} f(x) = f(c)$). Assume $c$ is a cluster point. ...
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votes
2answers
42 views
Understanding proof of sequential continuity?
I'm trying to understand proof of the following statement:
Q. Let $f$ be a function on a closed bounded interval $[a,b]$. Prove that $f$ is continuous at $ c \in [a,b]$ if and only if $f(x_n) \to c$...
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votes
0answers
32 views
How do I prove using an $\epsilon - \delta$ proof that $\lim_{x\rightarrow \frac{1}{e}}(e^{x^{x^x}})<2$?
Not a homework question. Just wanting to refresh my epsilon delta proofs, and came up with this - struggled for an hour, no idea where to start.
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votes
3answers
51 views
Using the $\varepsilon − N$ definition of the limit, prove that $\lim \limits_{n\to\infty} \frac{(n^2 + 1)}{ (n^2 + 2)} = 1$.
Using the $ε − N$ definition of the limit, prove that
$\displaystyle\lim \limits_{n\to\infty} \frac{(n^2 + 1)}{ (n^2 + 2)} = 1$.
In other words, given $\varepsilon> 0$, find explicitly a natural ...
0
votes
2answers
47 views
Question about the proof of the $\epsilon-\delta$ definition of continuity
I am currently trying to get my head around the proof of the definition of continuity of a function given in my Elementary Analysis textbook. The definition given is:
Let $f$ be a real-valued function ...
0
votes
1answer
42 views
Intuitive explanation of sign preservation of limits and boundedness?
Is there an intuitive explanation of the following two statements?
Let $I \subseteq \mathbb{R}$ be an open interval, let $c \in I$ and let $f:I - \left\{c\right\} \to \mathbb{R}$ be a function. If ...
-1
votes
1answer
32 views
proof lim x-a f(x)= lim x-0 f(x+a) ( duplicate)
i guess the proof here(Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$) is something wrong, its too easy and i thought i couldnt change things like the one most voted
0
votes
1answer
89 views
Whether the product of uniformly continuous functions is uniformly continuous [closed]
I know it isn't and I have to give a counter-example.
Function $f_1(x)=f_2(x)=x$
this is a uniformly continuous function
the product of these functions $f_1(x)\cdot f_2(x)=x\cdot x=x^2$ this isn't an ...
