Questions tagged [epsilon-delta]

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

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25 views

Prove Limit $5/(x+1)^3$ as $x \to -1^{-}$ is $-\infty$ epsilon delta proof

I have to prove the following using the epsilon delta method $$ \lim_{x \to \,-1^{-}} \, \frac{5}{(x+1)^3} = - \infty $$ So, using the language of quantifiers, I think, I have to prove the following $$...
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1answer
13 views

Given the function,check the continuity at $(0,0)$ and find $f_x(0,0),f_y(0,0)$ if they exist

Given a function defined by : $$f(x,y) = \begin{cases} \frac{x^{3}+y^{3}}{x^{2}+y^{2}} & (x,y) \ne (0,0) \\ 0 & (x,y) = (0,0) \end{cases}$$ Find $f_x(0,...
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84 views

$\forall\varepsilon > 0\exists\delta > 0:(|f(x) - \ell| < \varepsilon)\implies(0<|x-a| < \delta )$. $f(x)=x$ satisfies this for all $a$ and $\ell$?

This is to be found in the answer to problem 26 (b) of chapter 5 of Michael Spivak's Calculus 3rd edition. The problem reads as follows: 26 Give examples to show that the following definitions of $\...
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3answers
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What is wrong in this choice of $\delta$?

Suppose I have a wrongly evaluated limit$-$ $$\lim_{x \to2} 3x+3=6$$Then we get $-$ $$3x+3-6<\varepsilon $$ $$3x-3<\varepsilon $$$$|x-1|<\frac\varepsilon {3}$$$$|x-2|<\frac\varepsilon 3 -...
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2answers
65 views

Prove $\lim\limits_{x \to \infty} x \sin^2 x$ does not exist.

I need to prove $\lim\limits_{x \to \infty} x \sin^2 x$ does not exist, and request your help verifying my argument is correct. I've got somewhat decent at making $\epsilon-\delta$ arguments, but ...
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3answers
65 views

What's wrong in this $\epsilon-\delta$ argument?

In Spivak's Calculus, one exercise asks whether the following is true: Let $f$ and $g$ be functions such that $f(x) < g(x)$, for all $x$. Does it follow that $\lim\limits_{x \to a} f(x) < \lim\...
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5answers
54 views

How to prove that $f(x, y)=\frac{2xy}{x^2 + y^2}$ is not continuous in $(0, 0)$ using an Epsilon-Delta proof?

In an assignment for the course Real-Analysis I need to proof that $f(x, y)=\frac{2xy}{x^2 + y^2}$ is not continuous in $(0, 0)$ using an Epsilon-Delta proof. However, me and my fellow students have ...
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1answer
61 views

$\lim_{z\to\infty} \frac{4z^2}{(z-1)^2} = 4$

So, I'm having trouble performing the $\varepsilon$ - $\delta$ for this proof. I made the change from $z$ to $\frac{1}{z}$ so the limit becomes $\lim_{z \to 0} \frac{4}{(1-z)^2} = 4$. Next, I applied ...
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41 views

How to show that $x, y$ $\in$ $\mathbb{R}$ exist such that $f(x) < 0 < f(y)$ using a Epsilon-Delta proof?

For a small part of a school assignment (real Analysis) I have to show that $x, y$ $\in$ $\mathbb{R}$ exist such that $f(x) < 0 < f(y)$ Let $f$: $\mathbb{R}$ $\longrightarrow$ $\mathbb{R}$ ...
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1answer
85 views

Prove $f(x) = x^2 + 3$ is continuous at $x=3$

Prove that $f(x) = x^2 + 3$ is continuous at $x=3$. I have tried using $\delta = \sqrt{\epsilon + 9} - 3$. I tried to split $|x^2-9| = |(x-3)(x+3)|$ and tried to make $x+3$ in terms of $\delta$. But I ...
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1answer
56 views

Show $\cosh x$ and $\sinh x$ are continious using $\varepsilon - \delta$ proof

I have to prove that $\sinh x$ and $\cosh x$ are continuous functions. I have to use the hyperbolic addition formula, and the inequalities: $|\sinh x| \leq 3|x|, \, |x|<\frac{1}{2}$ $|\cosh x -1| ...
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1answer
32 views

Conditions on asymptotes of counting-functions

Definitions Let $M$ be an infinite subset of $\mathbb{N}$. For $x\ge0$ we define $$\pi_M(x) = \sum_{n\in M \leq x}1.$$ Also let $m_n$ denote the $n$-th element of $M$. What I was able to solve Let $f_{...
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2answers
92 views

Formal proof for the limit of $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$

Formal proof for the limit of $\frac{\tanh(x)-1}{e^{-2x}}$ as $x \rightarrow \infty$. So far Keep in mind I have to use the definition for a limit. I.e for this would be a proof for the limit at $\...
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3answers
71 views

How to show that $\lim_{(x,y) \to (0,0)}\frac{x^{2}y^{2}}{\left|x\right|^{3}+\left|y\right|^{3}}=0$

How to show that $$\lim_{(x,y) \to (0,0)}\frac{x^{2}y^{2}}{\left|x\right|^{3}+\left|y\right|^{3}}=0$$ I know that $$\left|\frac{x^{2}y^{2}}{\left|x\right|^{3}+\left|y\right|^{3}}\right|\le\frac{x^{2}y^...
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1answer
32 views

$\lim_{x \to 0 }|x| = 0$

Given $\epsilon > 0$ we want to $\delta > 0$ such as $|x - 0| < \delta \Rightarrow |f(x) - f(0)| < \epsilon $. How \begin{align} |x| = \begin{cases} x, \text{ if } x > 0\\ -x, \text{ ...
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2answers
77 views

Limit of $(x^3 - 3x^2 + 2x)$ as $x \to 1$ using epsilon delta definition

I need to prove the following limit using epsilon delta method. I have come up this on my own just to practice skills. $$\lim \limits_{x \to 1} \,(x^3 - 3x^2 + 2x) = 0 $$ So, I need to come up with ...
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2answers
81 views

$\epsilon$-proof for the limit $f(x)=\frac{1}{\cosh{(x)}}+\log{\left ( \frac{\cosh{(x)}}{1+\cosh{(x)}} \right )}$ as $x$ goes to $\pm \infty$

I want to use a $\epsilon,\delta$-proof for the existence and value for the limit of $$f(x)=\frac{1}{\cosh{(x)}}+\log{\left ( \frac{\cosh{(x)}}{1+\cosh{(x)}} \right )}$$ for $x \rightarrow \pm\infty$. ...
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1answer
94 views

$\epsilon,\delta$-proof for the limit of $\log \sinh{(x^2)}-x^2$ for $x \rightarrow \infty$

I want to use a $\epsilon,\delta$-proof for the existence and value for the limit of $$\log \sinh{(x^2)}-x^2$$ for $x \rightarrow \infty$. Now, I know the definition for such proof to be $\forall \...
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1answer
43 views

Use epsilon-delta definition of limits to show that $\lim_{(x,y) \to (0,0)}(\sqrt{-x^2-y^2})=0$

Use epsilon-delta definition of limits to show that $$\lim_{(x,y) \to (0,0)}(\sqrt{-x^2-y^2})=0$$ We need to show that $$\forall \epsilon >0( \exists \delta >0( \forall (x,y) \in \mathbb R^2 (0&...
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1answer
37 views

Use epsilon-delta definition of limits to show that $\lim_{(x,y) \to (1,2)}(x^2+3x-4y)=-4$ and $\lim_{(x,y) \to (1,-1)}(x^2+y^2)=2$

Use epsilon-delta definition of limits to show that $$\lim_{(x,y) \to (1,2)}(x^2+3x-4y)=-4$$ $$\lim_{(x,y) \to (1,-1)}(x^2+y^2)=2$$ We need to show that $$\forall \epsilon >0( \exists \delta >0( ...
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2answers
50 views

Proof limit value with $\epsilon/ \delta$

I have to find the limit value for $f(x)=\sqrt{1+x}$ for $x \rightarrow0$. And then show with $\epsilon /\delta$ that I have found the right limit value. I have found the limit value to $\sqrt{1+x} \...
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2answers
57 views

Epsilon delta proof sketch of $\lim_{x\to1}\frac{1-\sqrt{x}}{1-x}=\frac{1}{2}$

I need to prove that $\lim_{x\to1}\frac{1-\sqrt{x}}{1-x}=\frac{1}{2}$. My first step was re-writing this as: $\frac{1}{1+\sqrt{x}}$. So now for the sketch of the proof i got: Let $\epsilon>0$. Note ...
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1answer
20 views

Use epsilon-delta definition of limits to show that $\lim_{(x,y) \to (1,2)}(3x+2y-1)=6$ and $\lim_{(x,y) \to (0,0)}(x^2+y^2)=0$

Use epsilon-delta definition of limits to show that $$\lim_{(x,y) \to (1,2)}(3x+2y-1)=6$$ $$\lim_{(x,y) \to (0,0)}(x^2+y^2)=0$$ We need to show that $$\forall \epsilon >0( \exists \delta >0( \...
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2answers
67 views

$\epsilon , \delta$-proof and choosing correct $\delta$

Let $$f(x)=\sqrt{1+x}$$ Show that if $x \rightarrow 0$ a limit value does exist. Furtheremore, find the limit value and explain the choice of $\delta$ w.r.t $\epsilon$ when the definitions of a limit ...
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1answer
40 views

Show existence and find the value of the limit

Show that $$\frac{\tanh(x)-1}{e^{-2x}}$$ has a limit for $x \rightarrow \infty$ and the method to finding it. I am guessing I have to use epsilon-delta-definitions for limits and then find a ...
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2answers
71 views

How to give an Epsilon-Delta proof of this limit $ \lim_{x\to 1} \frac{1-\sqrt{x}}{1-x} = \frac{1}{2} $? [closed]

How to give an Epsilon Delta proof of the following limit: $$ \lim_{x\to 1} \frac{1-\sqrt{x}}{1-x} = \frac{1}{2} $$ Thanks for your help in advance!
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2answers
58 views

Question regarding $\varepsilon$-$\delta$

Consider the following, I try to prove a limit, and get $\frac \varepsilon 3$=$\delta$. So, the length $\delta$ $<$ $\varepsilon$. My question is how will every $\varepsilon$ correspond with a ...
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1answer
68 views

My first proof that epsilon delta $\displaystyle\lim_{x \to 5}3x+2=17$

Prove: $\displaystyle\lim_{x \to 5}3x+2=17$ General format $\displaystyle\lim_{x \to a}f(x)=L$ (included to verify various parameters) What needs to be proved: $\forall\epsilon>0, \exists\delta>...
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2answers
67 views

The gist of the epsilon delta definition of limit

I am currently re-reading and re-learning with basic concepts in calculus/analysis. I have tried to prove certain limits exist using the epsilon delta definition. Many of these examples are fairly ...
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1answer
36 views

Prove the space curve doest not have an asymptote

Prove that the function $\mathbb{R} \to \mathbb{R}^2: t \mapsto \left(t, at^2 + bt +c\right)$ doesn’t have an asymptote. Here what I thought so far: First of all Im using this definition for a line $L$...
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1answer
30 views

Proof that square root of a sequence converges to the square root of the limit of the sequence.

I've seen other posts around here regarding this issue as well, so I apologise for the repetition. But I need help regarding a specific part of the proof that I don't see covered elsewhere. I was ...
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0answers
22 views

Why can Epsilon equal the absolute value of the function divided by $2$? ($\varepsilon = \frac{|f(\text{point of interest})|} 2 > 0$)

I am trying to solve a tricky textbook problem that requires the use of the epsilon-delta definition of a limit. However, with my elementary understanding of how to answer these kinds of problems, the ...
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1answer
17 views

Proving non-uniform-continuity for f,g when g(x) > f(x) for all x

Suppose $f, g$ are continuous functions on $\mathbb{R}$, s.t $f$ is not uniformly continuous on $\mathbb{R}$ and $g(x) > f(x)$ for all x. Is g not uniformly continuous? This seems true, and I'm ...
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1answer
36 views

Prove that for $ f: \mathbb R \to \mathbb R , f(x)=\frac {x-1} {x^2 + x + 2}, \lim _{x \to 0} f(x) =- \frac12 $ by epsilon delta

As you can see from the image, I am not too sure where to go from after this. Could anyone give me a (not too ambiguous) hint?
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1answer
89 views

Spivak Calculus Chapter 5 Problem 10 (b)

I am unsure about how to solve this problem and unfortunately I do not understand the given solution. The problem goes as follows (Spivak, Calculus, 3rd edition, pg. 107, Problem 10-(b)): Prove that ...
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2answers
75 views

Can anyone explain this to me? (Epsilon and Delta)

If I have the statement $\lim_{x\to 0}f(x)=0$ and $$ f(x)=\begin{cases} x,&x>0\\ x-0.3,&x<0 \end{cases} $$ how do I prove that the statement is false? Is this correct? $[f(x)-0]$ < e ...
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1answer
36 views

Implied By/Implies Continuity Using the Definition of Continuity

I have to determine which of some definitions are implied by or imply the continuity of a function in a point. So I guess I should play with $\varepsilon$ and $\delta$ one time I understood what the ...
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1answer
39 views

(Delta-Epsilon) How do i prove this?

So i am trying to practice questions about Delta-Epsilon and i got stuck on this one here, $x^2-8x+14=7$ as x approaches 1. i know that $|x-1|$<$d$ implies that: $x^2-8x+14=7$ $x^2 -8x+7$ $(x-1)(x-...
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1answer
80 views

How do i prove this using precise definition of a limit?

$(x^2 - 6x +11) = 6$ lim approaching 1 Given ϵ > 0 if |x−1|<δ then $|x^2 − 6x + 11 -6|$=$|x^2 - 6x +5|$<ϵ this gets factored to: $|x^2 - 6x +5|$ = (x-1)(x-5)<ϵ This is where i am stuck, ...
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1answer
27 views

Difference in calculating continuity and uniform continuity

I've done some exercises on continuity using the $\varepsilon$-$\delta$ definition. Now going through my script I saw that there is a stricter version of continuity called uniform continuity. After ...
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2answers
66 views

Prove $\lim_{ (x,y) \to (0,0)} \frac{x^3-y^3}{x^2+y^2}=0$ [closed]

Can some one prove the limit using epsilon delta method to prove that the limit exists $$ \lim_{(x,y) \to (0,0)} \frac{x^3-y^3}{x^2+y^2}=0 $$
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1answer
16 views

Uniform continuity choosing delta problem

I am trying to show that $x^2$ is uniformly continuous on the set union of intervals [n,n+$n^{-2}$] for all positive integer n. I have also checked the previous posts. But my question is that for a ...
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1answer
38 views

A question about $\varepsilon$-$\delta$ definition of continuity

How can I prove the $f(x)=x^{\frac{2}{3}}-1$ is continuous on $[ -1, 1 ]$. First I wrote the expression for continuity on $\varepsilon$-$\delta$ language. So, $\forall \varepsilon>0$, $\exists \...
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1answer
28 views

epsilon delta limit proof of $\lim_{x \to 0} \arctan{(x)}=0$

Let $\lim x→0 \arctan{(x)}=0$ then there exists $\delta>0$ such that if $0<|x|<\delta$ then $|\arctan{(x)}|< \varepsilon $ I know that, $|\arctan{(x)}|<\pi/2$ But how do I relate this ...
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0answers
10 views

Trying to find the $\delta$ in epsilon-delta continuity proof. [duplicate]

I am trying to prove the following function is continuous for all irrationals: $f(x) = \begin{cases} 0, & \text{if $x$ is irrational} \\ 1/n, & \text{if $x = m/n$} \end{cases}$ The question ...
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2answers
84 views

Proof Check: $4x^2+3x+17$ is continuous by using a $\epsilon$, $\delta$ Argument

Can anyone double check my following epsilon delta proof. I want to prove that the following function is continuous with an Epsilon Delta Argument. $$ f: x \in \mathbb R \mapsto (4x^2+3x+17) \in \...
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3answers
71 views

Thoughts about the use of $\epsilon$-$\delta$ language in proofs involving limits

The limit of a function $f: \mathbb{R} \to \mathbb{R}$ at $x_{0}$ is defined as follows: $\lim_{x \to x_{0}} f\left(x\right) = L \iff \forall \epsilon > 0, \exists \delta > 0, \forall x, 0 < ...
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1answer
34 views

epsilon delta limit proof verification of $\lim_{x \to 0} \frac{x-1}{x^2-1}=1$

First at all, Let $\lim x→0 \frac{x-1}{x^2-1}=1$ then there exists $\delta>0$ such that if $0<|x|<\delta$ then $|f(x)-1|< \varepsilon $ Proof: $\bigg|\frac{x-1}{x^2-1}-1\bigg|=\bigg|\...
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2answers
72 views

Prove that $(x^2+2x+1)$ is continuous by using a $\epsilon$ - $\delta$ argument

I want to prove that the following function is continuous with an Epsilon Delta Argument. $$ f: \mathbb R \rightarrow \mathbb R \; (x^2+2x+1)$$ So I started with $$\left\lvert f(x)-f(y) \right\rvert= ...
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0answers
54 views

Proof for Spivak 7-13b.

I just wanted to ask if this proof can be used for exercise 7-13b in Spivak Calculus The exercise: Suppose that $f$ satisfies the conclusion of the Intermediate Value Theorem, and that $f$ takes on ...

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