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Questions tagged [epsilon-delta]

For questions regarding $\varepsilon$-$\delta$ definitions of limits and continuity.

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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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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 ...
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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) - ...
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2answers
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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 ...
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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\...
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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?
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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: $$\...
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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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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 $...
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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 ...
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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 ...
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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 ...
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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 $\...
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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....
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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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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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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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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 $\...
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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 ...
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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 ...
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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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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| + ...
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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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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 $|\frac{\ln(f(x,y))}{f(x,y)}|$?
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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| < \...
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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)...
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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
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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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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 ...
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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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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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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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1answer
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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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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 ...
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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. ...
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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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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!
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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 ...
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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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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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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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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 ...
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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 ...
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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 ...
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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
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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 ...