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 (limits-colimits) instead.
41,756
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Maximise area of rectangle with fixed perimeter
I've got a problem where a rectangle's area must be maximised given a fixed perimeter of $60$m.
Assuming a length of $x$ and height of $y$ I wrote an equation $y = 30x - x^2$ which i differentiated, ...
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On the commutativity of integral and limit
Suppose that $f\in C^1$
I try to prove that $$\frac{d}{dt}\int_{0}^{t}h(t,x)dx=h(t,t)+\int_{0}^{t}\frac{\partial h}{\partial t}(t,x)dx.$$
However this depends on the fact that $$\lim_{\Delta t\to0} \...
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How to solve the limit of $\lim_{n\to\infty}\frac{n^{n + 1/2}}{n!e^{n}}$ as $n$ goes to infinity?
I found out from wolfram alpha that
$$\lim_{n\to\infty}\frac{n^{n + 1/2}}{n!e^{n}}=\frac{1}{\sqrt{2\pi}}$$
However, I don't know how to get to this answer.
I've tried l'hopitals, but that doesn't ...
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is it mathematically incorrect? [closed]
pls check if my solution to a physics question is mathematically correct or not
The link above contains the photo of my solution.
The question is:
A mass M is to be divided into two parts, m and (M-m) ...
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Can I split this limit into two like this? [closed]
Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$, and $f \in C^{1}$ (continuous and derivative continuous). Let $p,e,v \in \mathbb{R}^{n}$.
This is supposed to be true (is part of the proof of ...
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Expanding two iterated limits
I have always had a little trouble fully understanding iterated limits. So I thought it might be a good exercise to expand it in quantifiers.
Suppose we want to expand $\displaystyle \lim_{x \to a} f'(...
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Limit of the sum of cosine
I want to find the value of $$\sum_{k=0}^\infty \cos\left[\left(k+\frac{1}{2}\right)\pi x\right]\cos\left[\left(k+\frac{1}{2}\right)\pi t\right].$$
I think it should relate to delta function because ...
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This question I have solved in 2 ways : the first one gives me the answer "0" and the second one gave me the answer "ln(a) x ln(b)" [closed]
The Question is :
Please figure out what is the problem and where I made a mistake (if there are any). By the way, the correct answer according to the book is ln(a) x ln(b), which is the 2nd ...
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1
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Doubts in $\lim_{x \to +\infty} \frac{\int_{-x}^x \frac{1}{y^2}dy}{x}$
Consider the limit:
$$\lim_{x \to +\infty} \frac{\int_{-x}^x \frac{1}{y^2}dy}{x}$$
Since $\int_{-x}^x \frac{1}{y^2}dy=\int_0^x \frac{1}{y^2}dy-\int_0^{-x} \frac{1}{y^2}dy$, using Hopital's rule:
$$\...
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There is $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ with $f(0) = a$ and $f'(0)= b$
Suppose that $a,b \in \mathbb C \backslash \{0\}, a \neq 1$. Prove that there is some $R > 0$ such that there are no holomorphic functions $f: \{z: |z| < R \} \to \mathbb C \backslash \{0,1\}$ ...
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What is the concept of a limit in simple terms
I'm 14 just starting high school and I got interested in all the new math so I took a little dive into calculus but I can't wrap my head around the concept of a limit and for which uses it is applied ...
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Limit Derivation and Proof (Rigorous ε-δ Proof) [closed]
Could someone, please, provide me with a rigorous ε-δ proof for the following:
$$
\lim_{x \rightarrow 4^{-}}\frac{\sqrt{x}-1}{(x^{2}-16)^{5}}=-\infty
$$
?
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Prove limit of a function that goes from $\mathbb{R}^{2} \rightarrow \mathbb{R}$
I need to find $\lim_{(x,y) \to (0,0)} f(x,y)$ where $f(x,y) = \frac{\sqrt{(1 + 4x)(1 + 6y)}-1}{2x + 3y}$. I have already computed such limit by fixing $y = 0$, and it turned out it went to 1, and ...
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Does this limit exist and if it does what is it? [closed]
I was playing around with a few values of $k$ between $0$ and $1$ on Wolfram alpha and found that for all of them
$$\sum_{n= 1}^\infty \frac{\sin(n)}{n^k}$$
converges and I was wondering about the ...
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Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
Does the Distributive Property of Multiplication Over Addition apply to Absolutely Convergent Infinite Sums
For example, is it true that $\sum\limits_{n=1}^\infty (k\cdot a_n) = k\left(\sum\limits_{n=...
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How to evaluate $\lim_{k\to\infty}\frac{k}{3^{k}}$ [duplicate]
I have to evaluate
$$\lim_{k\to\infty}\frac{k}{3^{k}}$$
At first, I rewrote it as
$$\lim_{k\to\infty}\frac{3^{-k}}{\dfrac{1}{k}}$$
and then, applying L'Hopital rule, I got
$$\lim_{k\to\infty}\frac{\ln(...
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How to prove the limits about integral. [closed]
Let $f (x)$ be continuous on $[0,a](0<a<\pi)$.prove that
$$\lim_{\lambda\to+\infty}\frac{1}{\lambda}\int_{0}^{a}f(x)\frac{\sin^2\lambda x}{\sin^2x}\mathrm{d}x = \frac{\pi}{2}f(0).$$
I thought ...
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Does division by 0 really have anything to do with $\lim_{x \to 0} \frac{1}{x}$?
To my very limited knowledge, division by 0 is undefined precisely because it breaks the field axioms. No dividing by 0 if you want a field. However there do exist structures that are not fields which ...
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About the theorem that links limits of sequences and limits of functions, and the arbitrary delta.
First things first, I apologize if at any point I spell any formulas incorrectly or if at any point I name something wrongly (I am taking the class in another language and I am translating everything ...
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Limit of $\sum_{t=0}^n a_n(t)$ for n tends to infinity
I am wondering about how to deal with a limit of the following form
$$ \lim_{n \rightarrow \infty} \sum_{t=0}^n a_n(t),$$
with the function $a_n(t)$ and the sum being dependent on $n$.
Consider the ...
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Is this definition of infinite limits needlessly restrictive?
I am studying James Stewart's "Calculus: Early Transcendentals 7th Edition". On page 115, a precise definition of infinite limits is presented:
Let $f$ be a function defined on some open ...
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Find $\lim_{x\to0}\frac{\arctan\left(e^x-1\right)-e^{\arctan(x)}+1}{x^4}.$ [closed]
Evaluate the following limit:
$$\lim_{x\to0}\frac{\arctan\left(e^x-1\right)-e^{\arctan(x)}+1}{x^4}.$$
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Find the limit $\lim_{x\to 0}\frac{\sin(x)\sin^{-1}(x)-\sinh(x)\sinh^{-1}(x)}{x^2(\cos(x)-\cosh(x)+\sec(x)-\text{sech}(x))}$
Here is an example from the book "Asymptotic Analysis and Perturbation Theory" by William Paulsen.
Example 1.10 p.18.
Find $$\lim_{x\to 0}\frac{\sin(x)\sin^{-1}(x)-\sinh(x)\sinh^{-1}(x)}{x^...
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Solve $\lim\limits_{x\rightarrow 0}\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}$
$\displaystyle {\lim _{x\rightarrow 0}{\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt[3]{1+x}-\sqrt[3]{1-x}}}=\lim _{x\rightarrow 0}{\frac{(\sqrt[3]{1+x}-\sqrt[3]{1-x})((\sqrt[3]{1+x})^2+(\sqrt[3]{1+x})(\sqrt[3]{...
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Proof of convergence of $~\sum_{n=1}^{\infty}\frac{n^2+7}{2n^4-n+3}~$
I want to prove that the following infinite series converges.
$$\begin{align}
A:&=\sum_{n=1}^{\infty}\frac{n^2+7}{2n^4-n+3}\\&=\sum_{n=1}^{\infty}\frac{1}{n^2}\underbrace{\left(\frac{1+\frac{7}...
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How to solve the following limit
If
$$\lim_{x\to0}\frac1{x^m}\prod_{k=1}^n \int_0^x\big[k-\cos(kt)\big]\mathrm dt$$
exists and is equal to $20$ (where $m,n\in\mathbb N$) then what is the value of $n$?
I started this question with ...
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How to evaluate the following summation?
I have managed to solve the first half of the question by using A=kπ/2n and B=(k-1)π/2n, furthermore,How do you solve the second part of the question?
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Does uniform convergence on any interval of form $[-M,M]$ imply continuïty of limit function?
For the past few days I've been studying Analysis and more specifcally the topic of continuïty. I was making some exercices on this topic and got stuck with a problem which goes as follows. Let $(f_n)...
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For $a\in\mathbb{R}$, let $f_a:\mathbb{R}\to\mathbb{R}, f_a(x) = \int_{-x}^x \frac{e^t\cdot\cos(at)}{1+e^t}dt$, determine $\lim_{a\to0}f_a(x)$.
For $a\in\mathbb{R}$, let $$f_a:\mathbb{R}\to\mathbb{R}, f_a(x) = \int_{-x}^x \frac{e^t\cdot\cos(at)}{1+e^t}dt,$$ and $L=\lim_{a\to0}f_a(x)$. Calculate $L$.
I am completely lost with this problem. My ...
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In this examples limit exist or not, please justify? [closed]
$a)$
$$ \lim_{n \to \infty}\left(\frac{\sin \sqrt{n}}{\sqrt{n}}\right)$$
$b)$
$$\lim_{n \to \infty}\left(\frac{3n^2-5}{3n^2-1}\right)^{\sqrt{n^5}} $$
how to find the limit when sin has sqrt and in the ...
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The limit $\lim\limits_{n\to\infty}\frac{x^n}{n!}$
Evaluate $$\lim_{n\to\infty}\frac{x^n}{n!}$$ where $x\in\mathbb R$
This is a common limit and has been asked and answered many times here on this site. However, I present another approach with Stolz-...
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Weak convergence of functionals $g_n^*(f) = n\int_0^1 x^nf(x)dx$ [closed]
Show that sequnce of functionals $g_n^*(f) = n\displaystyle{\int_0^1 x^nf(x)dx}, f \in C[0,1]$ converges weakly and find its limit functional. Does it converge in the norm of space $C^*[0,1]$?
I don'...
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When can you use the average value of a function on $[a,b]$ for limits?
My teacher has taught in class the following theorem, which I will write down, as I do not know the English name and don't want to create any confusion:
Let $f:[a,b]\to\mathbb{R},\ f$ continuous. ...
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Wrong result when calculating the answer with an infinite series
Two trains, each having a speed of 30km/h, are headed at each other on the same straight track. A bird that can fly 60km/h flies off the front of one train when they are 60km apart and heads directly ...
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Finding Polynomials of Lowest Degree that is similar to $f$ as $x\to a$
I'm working through the book "Asymptotic Analysis and Perturbation Theory" by William Paulsen and am having some trouble with the following problems.
Definition 1.1 p.1. Given two functions,...
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Is every permutation $\varphi:\mathbb{N}\to\mathbb{N}$ well-balanced? [closed]
Let $\mathbb{N}$ denote the set of positive integers. For any function $f:\mathbb{N}\to\mathbb{N}$ we let $$d(f) = \lim\inf_{n\to\infty}\frac{\sum_{i\leq n}f(i)}{\sum_{i\leq n}i}\text{ and } D(f) = \...
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a function defined by inferior limit related to Borel measure is Borel measurable
Question:
$\mu$ is a Borel measure on $\mathbb R$.
Define $f:\mathbb{R\to \bar R},f(x)={\operatorname{lim inf}}_{r\to 0}{{\mu((x-r,x+r))}\over{r}}$.
Prove that $f$ is (extended) Borel measurable.
I ...
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Are there other, non-probabilistic ways to calculate: $\lim_{n\to\infty}\frac{1}{n}\ln\sum_{m>n\alpha}\frac{(n\lambda)^m}{m!}$?
In section five of this nice exposition of moment generating functions, we prove the following theorem:
Take an i.i.d sequence of random variables $(X_n:\Omega\to\Bbb R)_{n\in\Bbb N}$ whose common ...
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Find the limit $\lim_{n→∞} \frac{n+k}{n^2} \sum_{k=1}^n (\ln(n+k) - \ln(n))$ [closed]
My exercise is:
$$
\lim_{n→∞} \sum_{k=1}^n \frac{n+k}{n^2}(\ln(n+k) - \ln(n))
$$
I don't know how to solve it? Can you give me advices?
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How to prove limit of 1/x as x approaches a is 1/a [closed]
To prove limit of 1/x as x approaches the variable a is 1/a using the epsilon delta definition , i can prove the limit for any given number but when its a variable i get stuck since i dont know what ...
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Prove that if $\lim_{x \to \infty} f(x) = \infty$, $\lim_{x \to \infty} f(x + a) = \infty$ for any $a > 0$.
Hypothesis
Assume that $\lim_{x \to \infty} f(x) = \infty$ and $f'(x) > 0$ for all $x \in (d, \infty)$ ($d \in \mathbb{R}$), then $$\lim_{x \to \infty} f(x + a) = \infty.$$
Proof
Since $$\lim_{x \...
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Defining limits with sequences [closed]
So I saw a proof that $f(x)$ goes to $l$ as $x$ goes to $a$ if and only if for all sequences $x_n$ that converge to $a$, $f(x_n)$ converges to $l$. Here it is: https://math.stackexchange.com/a/2663864/...
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Does a continuous function of a sequence with a convergent Cesaro mean have a convergent Cesaro mean?
Suppose we have a sequence of real numbers $x_1, x_2, \dots$ whose Cesaro mean converges. That is, suppose there is a real value $\overline{x}$ such that
\begin{equation}
\lim_{n\to \infty} \sum_{i=1}^...
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3
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Prove that $\frac{\sin(xy + y^3)}{x^2 + y^2}$ has no limit at the origin
I have this function, $$ \frac{\sin(xy + y^3)}{x^2 + y^2} $$
I have to prove that for $(x, y)\to(0, 0)$, limit doesn't exist, and because of that, function is not continuous (i.e. function diverges).
...
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Need help figuring out a couple of steps behind equation logic
The problem describes the logic behind limits in math. f1
Based on an area under a curve. The overall idea is pretty clear. We split the OX line into several parts with the following coordinates f2
...
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Justification for a substituion that turns a finite sum to infinite - constructing the Grunwald-Letnikov fractional derivative (Fractional Calculus)
Steps in question
These steps raise numerous questions.
What is the reasoning behind choosing $\delta _Nx\equiv [x-a]/N$ ?
This seems almost arbitrary. I understand that $a$ and $x$ eventually ...
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sequences of random variables
enter image description here
My thought is find the characteristic functions of this random variables, but I am not sure is it OK or not.
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Limit of nth root of polynomial series
Let $P(n)$ be a polynomial of positive degree. Show that, $$\lim_{n\to \infty} \sqrt[n]{ |P(n)|} =1 $$
Here is my attempt, but it's clunky and I'm not 100% sure it if works or not, but here goes (...
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1
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47
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The limit of a Nasty Summation
I'm trying to evaluate the limit as h approachs 0 of the sum from k = 0 to n of:
$\frac{1}{h^n}(-1)^{k+n}\binom{n}{k}\frac{1}{(x+kh)^2-2(x+kh)+17}$
If it helps, it's the limit definition of the nth ...
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2
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Is this statement correct : $\frac{dv}{dt}\times dx=\frac{dx}{dt}\times dv$
So I just came across a physics derivation where the treat the $dv$ and $dx$ operators like fractions while I have always heard it's a mistake. But so far what I came up with:
$$\begin{align*}
\frac{...
