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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Is my proof rigorous enough? Spivak, Prove:$\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$ (Chapter 5, Problem 10)
I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect).
Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$. (...
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Trouble Understanding Difference in Epsilon-Delta Arguments: Why One Works But The Other Fails (Spivak Calculus Problem 5-10c)
In the problems for the limits chapter (5) of Spivak's Calculus, we are asked to prove: $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)$. The relevant to the question proof alternative is:
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Help understanding Spivak's solution, and a verification of my proof. Spivak Chapter 5, Question 3(vi)
Find a $\delta$ such that if $0<|x-1|<\delta$ then $|\sqrt{x}-1|\lt
\epsilon$
My solution:
$|\sqrt{x}-1|\cdot |\sqrt{x}+1|=|x-1|$
$
\epsilon>|\sqrt{x}-1|\ge |\sqrt{x}|-1$ which implies $\...
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Understanding epsilon-delta proof regarding dense sets
I'm currently self-studying using Spivak's "Calculus" and I wanted to check on my understanding regarding an epsilon-delta proof for dense sets. The first problem was the following:
If $f$ ...
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Using Graph of $\frac{1}{x}$ to Find $\delta$ [closed]
Hello,
This is a question I was assigned for homework and I am struggling to find a correct answer. The question states "Use the graph of $f(x)=\frac{1}{x}$ below to find a number δ such that $|f(...
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If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following
If $\lim_{x\to a}f(x)=\infty$ and $\lim_{x\to a}g(x)=c$, where $c$ is a real number, prove the following:
$\lim_{x\to a}[f(x)+g(x)]=\infty$
$\lim_{x\to a}[f(x)g(x)]=\infty $ if $c>0$
$\lim_{x\to a}...
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Finding $\delta$ for convergence of $f(x)=x^3+2x^2-3x-1$ at its approximate root.
I've had some exposure to elementary analysis and I am currently going through a problem in numerical analysis involving finding roots using Newton's method. The algorithm has convergence issues which ...
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Sign permanence of locally Lipschitz functions calculated on a sequence
Suppose I have a sequence $a_k(m)>0$ with $m\in\mathbb{N}$ such that, given $k\in\mathbb{N}$ and $p\geq 1$, I can show that $$|a_{k+p}(m)-a_k(m) |\leq \frac{m^2}{k^2}$$ with $a_{k+p}<a_{k}$. I ...
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Proving that a function with only removable discontinuities can be made continuous
I'm working with Spivak's "Calculus" and was doing the following problem:
Let $f$ be a function with the property that every discontinuity is a removable discontinuity. This means that $\...
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The continuous functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ satisfy $f(x) = g(x)$ for all $x \in \mathbb{Q}$. [duplicate]
(a) Using the definition of continuity, prove that $f(x) = g(x)$ for all $x \in \mathbb{R}$. (b) Use sequential criteria of continuity to redo the problem.
I was able to do the part (b) of this ...
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On continuous functions and convergent sequence
The function $f : (0, 1] \to\mathbb R$ is a bounded and continuous on $(0, 1]$. Let $\{x_n\}$ be a sequence in $[0, 1]$. Prove or disprove the following.
(a) If $\{x_n\}$ is convergent, then $\{f(x_n)\...
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Conditions equivalent to the ε-N definition
I want to know the condition equivalent to the ε-N definition. I think TFAE. But, I can't prove [2. ⟹ 1.].
$(a_n)$ is a sequence such that $\forall n\in \mathbb{N},a_n\in \mathbb{R}$,and $I\subset \...
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Prove using epsilon - delta
let $f(x) = \dfrac {x-\sqrt x} {5x-4}$
prove using $\epsilon-\delta$ that $\lim_{x\to1} f(x) = 0$
I got it to -
let $\epsilon >0$.
choose $\delta = min(0.1 , \dfrac \epsilon 2)$
Than for all $\...
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Proof that $\lim_{x\to 0} f(x) = \lim_{x\to 0}f(x^3)$ [duplicate]
I wanted to prove the following theorem and was wondering if my line of reasoning was correct:
Theorem: $\lim_{x\to 0}f(x) = \lim_{x\to 0}f(x^3)$
Proof:
Suppose $\lim_{x\to 0}f(x)$ exists and equals $...
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Verification of an epsilon-delta proof
I'm following Michael Spivak's "Calculus". Here's the question:
$$0 < |x - 2| < \sin^2\left(\frac{\epsilon^2}{9}\right) + \epsilon\implies |f(x) - 2| < \epsilon$$
$$0 < |x-2| <...
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