Questions tagged [limits]
Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.
44,124
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Use $\,\varepsilon-\delta\,$ definition of limits to show that $\lim\limits_{(x,y)\to(-1,2)}\frac{x^3+y^3}{x^2+y^2}=\frac{7}{5}$
We need to show that for every $\varepsilon >0$ there exists a $\delta >0$ such that
$$\left\lvert\frac{x^3+y^3}{x^2+y^2}-\frac{7}{5}\right\rvert<\varepsilon $$
whenever
$$0<\sqrt{(x+1)^2+(...
- 117
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Is my proof rigorous enough? Spivak, Prove:$\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$ (Chapter 5, Problem 10)
I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect).
Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$. (...
- 135
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2
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Trouble Understanding Difference in Epsilon-Delta Arguments: Why One Works But The Other Fails (Spivak Calculus Problem 5-10c)
In the problems for the limits chapter (5) of Spivak's Calculus, we are asked to prove: $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)$. The relevant to the question proof alternative is:
...
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Growth of factorial functions and tetration
Good morning everyone, I have a doubt about the growth rate of the following two functions:
$$ \operatorname f(x)=x! $$
and
$$ \operatorname g(x)=^x n,\quad\mbox{where}\ \quad n \in \mathbb{R^+} $$
I ...
- 45
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votes
1
answer
45
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Limit of a function when 'a' is not in the domain [closed]
Is this a correct statement that as x approaches to 'a' for f(x) where 'a' does not belong to the domain of f(x) then the limit at 'a' does not exist
2
votes
1
answer
39
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Is the approach I did to justify this limit equality correct?
I am reading a solution to a problem and got stuck to understand the equality (if $A(t)$ is an invertible matrix whose entries depend in a differentiable way on a parameter $t$ and $ \lim_{h\to0}P(h)=...
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votes
0
answers
40
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Recurring A.M-G.M. in limiting case [closed]
I came across a question where a recurring AM-GM sequence converges in the limit, but I am not able to find point of convergence as a closed form expression.
Question: Let $f(x)$, $g(x)$ be defined as:...
0
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0
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48
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Does this recurrent sequence have a limit?
I have a sequence $a_1 = 1$, $a_2 = 0$, $$a_k = \frac{k*(k+2)+(k+1)}{k*(k+2)}(a_{k-1} -\frac{k-1}{k*(k-2) + (k-1)}a_{k-2})$$ for $k \geq 3$,
I want to know whether this sequence would converge to $0$ ...
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A limit involving the $n$th derivative of the reciprocal of the Riemann Zeta function: $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ [closed]
Does the limit $\lim _{n \to \infty} |(1/\zeta)^{(n)}(2)|$ exist? If so, how can we determine its precise value, or is there any way to approximate it with high precision? ($(1/\zeta)^{(n)}(2)$ is ...
- 43
2
votes
2
answers
74
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Limit of $\sum_{k=1}^n \frac{1}{k+1/2} - \ln(n+1/2)$
I am trying to understand the limit of the following expression as $n \to \infty$:
$$\sum_{k=1}^n \frac{1}{k+1/2} - \ln(n+1/2)$$
Numerical investigation suggests that the limit of this expression as $...
- 105
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votes
1
answer
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Newton approximation in Tao Analysis 1
I'm reading Tao's Analysis 1 and I'm confused about this. Help!
Definitions that are relevant:
Differentiability at a point:
Let $X$ be a subset of $\mathbb{R}$, and let $x_0 \in X$ be an element of $...
1
vote
1
answer
44
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Help understanding Spivak's solution, and a verification of my proof. Spivak Chapter 5, Question 3(vi)
Find a $\delta$ such that if $0<|x-1|<\delta$ then $|\sqrt{x}-1|\lt
\epsilon$
My solution:
$|\sqrt{x}-1|\cdot |\sqrt{x}+1|=|x-1|$
$
\epsilon>|\sqrt{x}-1|\ge |\sqrt{x}|-1$ which implies $\...
- 135
1
vote
1
answer
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$A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2 < 4z\}$ limit: $\lim_{n \to \infty}\frac{1}{n} \int_A \frac{y^2z}{ln(x^2+2y^2+n) - ln(n)} \ d\lambda_3$
$A= \{(x,y,z) \in \mathbb{R}^3:x^2+2y^2+z^2<4z\}$. Calculate the limit:
$$\lim_{n \to \infty}\frac{1}{n} \int_A \frac{y^2z}{ln(x^2+2y^2+n) - ln(n)} \ d\lambda_3$$
Solution:
First, I can parametrize ...
- 1,096
5
votes
1
answer
57
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How to show that the limit of a sequence is not equal to some value?
In the second chapter of the book Understanding Analysis by Abbott, Example 2.2.6 proved that picking $N>\frac{1}{\varepsilon}$ suffices to prove the convergence of $\frac{n+1}{n}$ to the number $1$...
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How to prove the following limit is 0? [closed]
How to prove that
$$
\lim_{x\to 0^-} \frac{e^{1/x}}{x} =0 ?
$$
