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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.

0
votes
1answer
25 views

Limit counterexample for infinite union

It is well known that if a function has the same limit on finitely many sets, then the limit exists on the union of those sets. (just choose the minimal delta). I am trying to find a counterexample to ...
0
votes
1answer
17 views

Relation between limit equals to infinite and limit doesn't exist. [on hold]

Can anyone please tell me the relation between limit equals to infinite and limit does not exist?
-1
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0answers
20 views

A taylor series for $\lim \limits_{h \to 0} \frac{\zeta'(-2n-h)}{\zeta(-2n-h)}$ [on hold]

Please help and use Big O Notation, thanks. I tried so-far making a Taylor Series for this limit but my solution is not what I seek for.
4
votes
1answer
45 views

$\{x_n\}$ is a bounded above sequence such that $x_{n+1} - x_n \ge a_n$, where $\sum a_k$ converges. Prove $x_n$ converges.

$\{x_n\}$ is a bounded above sequence satisfying the following property: $$ x_{n+1} - x_n \ge \alpha_n\tag1 $$ where $\alpha_n$ is such that $$ \exists \lim_{n\to\infty} \sum_{k=1}^n \alpha_k ...
0
votes
1answer
24 views

Given a bounded above sequence such that $x_{n+1}-x_n \ge -{1\over 2^n}$, prove that $x_n$ converges. [duplicate]

Prove that a bounded above sequence converges given it satisfies the following property: $$ x_{n+1}-x_n \ge -{1\over 2^n}\\ n\in\Bbb N $$ Since the sequence is bounded above, by definition we have:...
0
votes
0answers
34 views

A concrete question in Real Analysis

Suppose that $\{x^k\}$ is a set of $n$-dimensional vectors. $\omega\in\Omega$ and $\Omega$ is a measurable set. Denote $v_i$ as the $i$-th component of $v$ if $v$ is a vector. $\rho(\cdot)$ is a ...
-1
votes
0answers
19 views

Class 10 geometry [on hold]

What is method of limits which is used for proving various problems for class 10 geometry.it is being often asked in my book however this concept is supposedly different from the limits in calculus.
1
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4answers
52 views

Evaluate limit without using L'H rule

I was revising teacher's notes about L'H rule and I came across this limit. $$\lim_{x \to \frac{\pi}{2}} \frac{x-\frac{\pi}{2}}{\sqrt{1-\sin x}}$$ The teacher tried to evaluate without L'H to ...
0
votes
0answers
33 views

Help with Limit 9

I have the following system of partial derivatives: $$\frac{\partial Y}{\partial K}=\frac{1}{K}\left ( Y-\frac{\partial Y}{\partial L}L \right )$$ $$\frac{\partial Y}{\partial L}=\alpha \left (\frac{...
1
vote
3answers
42 views

Evaluate the following limit: $\lim_{x\downarrow 0}\dfrac{1}{x^c(1 - e^x)}$

I need to evaluate the following limit: $$ \lim_{x\downarrow 0} \dfrac{(1 - e^x)^{-1}}{x^c} $$ for different values of the constant $c$. What I've tried thus far: We have that $$ \lim_{x\downarrow 0}...
0
votes
0answers
22 views

What are the limit points of $X \times \mathbb{N}$?

Let $X=\{0,1,2 \}$ with natural order topology and give $\mathbb{N}$ its natural order topology. Consider $X \times \mathbb{N}$ with the dictionary order topology. What are the limit points of $X \...
2
votes
3answers
42 views

Use L'Hospital's rule to find $ \lim \limits_{x \to 0} \left( \frac{ \tan\beta x - \beta \tan x}{\sin \beta x - \beta \sin x} \right) $

Use L'Hopital's rule to find $ \lim \limits_{x \to 0} \left( \frac{ \tan\beta x - \beta \tan x}{\sin \beta x - \beta \sin x} \right) $ where $\beta $ is a non-zero constant and $\beta \ne \pm 1$. I ...
2
votes
4answers
61 views

Why is $x+e^{-x}>0$ for all $x \in \mathbb{R}$?

Denote $f(x) = x + e^{-x}$. Note that $f(0) = 1$ and $f'(x) = 1-e^{-x}$. That means $$\lim_{x\rightarrow -\infty} f'(x) = -\infty.$$ So if the rate of change of $f(x)$ keeps decreasing exponentially ...
0
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0answers
13 views

Notation: limit with no variable under it next to infimum with variable under it

My statistics text has the following definition for a uniform asymptotic confidence interval: $$\lim \inf_{n \to \infty} \inf_{\theta \in \Theta} \mathbb{P}(\theta \in C_n) \geq (1-\alpha)$$ How do ...
1
vote
1answer
22 views

For which value of $x$ is the average rate of change equal to the instantaneous rate of change?

The average rate of change for $f(x)=x^2+4x-6$ on the interval $[1,3]$ is $8$. I am not interested in final answer but more how to get there. I am going through calculus right now and already know ...
0
votes
3answers
38 views

Indeterminate multivariable limit [on hold]

How would I evaluate an indeterminate limit such as: $$\lim_{x,y \to 0} \frac{3x^2-3y}{3x-3y^2}$$ that is a function of two variables
0
votes
2answers
36 views

Evaluation of $\lim_{x\to\infty} \frac{1}{x}\int_1^x \frac{\log(1+s)}{(1+\log(s))\sqrt{1+s^2}}ds$ [on hold]

I really don't know how to start. I need an hint to get started. Thank you.
0
votes
1answer
24 views

Sequence of functions uniformly/absolute/normal convergence

I've got this sequence of functions... $f(x) = \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$ To show: The sequence is convergent in $\mathbb{R}$ It is NOT uniformly convergent in compact intervals [...
1
vote
4answers
49 views

Correct way to solve limit with square root in denominator

I would like to know how to correctly solve limit as $x$ approaches negative infinity for the following expression. $$\frac{2x+1}{\sqrt{4x^2-2}+1}$$ I would post my attempt at a solution but I ...
0
votes
0answers
21 views

Using the tool of asymptotic expansions to solve limit related questions in real analysis

I have seen that on math.stackexchange.com lot of people use asymptotic expansions to solve the problem of finding the limit of a sequence. I am reading from Ross's book on Analysis but he does not ...
2
votes
2answers
57 views

Computing $ \lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right]$ [duplicate]

$$ \lim _{n \rightarrow \infty}\left[n-\frac{n}{e}\left(1+\frac{1}{n}\right)^{n}\right] \text { equals }\_\_\_\_ $$ I tried to expand the term in power using binomial theorem but still could not ...
-2
votes
1answer
53 views

How can I know the derivability of this function?

I've been working on this problem: Study the differentiability of the following function $f$ at $x=0$: $$ f=\begin{cases} \dfrac{\cos(3x)e^{3x} - e^x}{\ln(1+x)} & \text{if} x>0\\ 2 &\text{...
0
votes
1answer
27 views

Prove that any bounded sequence may be split up in countably many sequences having the same limit.

Given a bounded sequence $\{x_n\}$ prove that it may be split into countably many sequences having the same limit. The question is stated as above, without any other constraints on the sequence. I'm ...
0
votes
0answers
8 views

Convergence of unit vector in dir. of $M^{k}*v$ to the principal eigenvector of M when $k\to \infty$ and M is symmetric

It is a standard fact that a square matrix $M$ of dimension $n$ has at most $n$ distinct eigenvalues, each of them a real number, and the sum of their multiplicities is exactly $n$. We will denote ...
2
votes
1answer
37 views

Is this geometric mean-like limit equal $0$?

Let $(a_n)_n$ be a sequence of positive real numbers such that $a_n\leq1$ for all $n\in\mathbb{N}$, and suppose that $\displaystyle\lim_{n\to\infty}a_n=0$. By the Stolz–Cesàro theorem we know that $\...
-1
votes
1answer
55 views

How to find $\lim\limits_{n\to \infty }\left(\frac{n}{1+n^2}+\frac{n}{4+n^2}+\frac{n}{9+n^2}+…+\frac{n}{2n^2}\right)$? [on hold]

How to find $\lim\limits_{n\to \infty }\left(\frac{n}{1+n^2}+\frac{n}{4+n^2}+\frac{n}{9+n^2}+...+\frac{n}{2n^2}\right)$? Any assistance with this problem would be appreciated.
-3
votes
0answers
18 views

elements of a bounded sequence all have a convergent subsequence [on hold]

Prove that if every element of a bounded sequence is part of a subsequence which converges to a, then the sequence converges to a.
-1
votes
5answers
39 views

How can I prove limit of $n^k$ over $c^n$ is 0?

How can I prove that $$ \lim_{n \to \infty}\frac{n^k}{c^n}=0\ ? $$ I know it is true by intuition, but I do not know how to prove it. Here $c\gt1, k\ge1$. BACKGROUND I am learning time ...
0
votes
2answers
31 views

Between proper integrals and improper integrals

I just started learning about improper integrals. Many of them are improper because the function evaluates to infinity at some point in their domains, e.g. $f(x)=1/x$ on the domain of $(0,1)$. My ...
0
votes
0answers
18 views

Geometric meaning of the Césaro limit of Geometric sequence on the Torus.

As a motivattion for an introdutory notion in our Ergodic lecuture, we were asked to give a Geometric meaning to the following limit. Let $\lambda\in \Bbb T$ $$\lim_{n\to \infty}\frac1n\sum_{k=0}^{...
3
votes
2answers
49 views

L'Hospital's Rule and indeterminate form $\frac{\infty}{-\infty}$

Suppose I have a limit of the form \begin{align*} \lim\limits_{x \to -\infty} \frac{x}{e^{x^2}}. \end{align*} As $x \to -\infty$, $x \to -\infty$ and $e^{x^2} \to \infty$. Now, if we were subtracting ...
1
vote
1answer
65 views

A tricky limit involving exponential integrals

We define exponential integral according to https://en.wikipedia.org/wiki/Exponential_integral#Definition_by_Ein as $$\text{Ei}_n(x) = \int_{1}^{\infty} \frac{e^{-xt}}{t^n} dt$$ I'm trying to ...
0
votes
0answers
58 views

how to prove that $\lim \limits_{x \to \infty}[f(x+1)-f(x)] = L \implies \lim \limits_{x \to \infty} [f(x)/x] = L$ [duplicate]

I don’t know how to formally prove that if $$\lim_{x \to \infty} \left(f(x+1)-f(x)\right) = L,$$ then $$\lim _{x \to \infty} \frac{f(x)}{x} = L$$ where L is a constant and the function is limited ...
0
votes
1answer
25 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. ...
3
votes
0answers
65 views

subset $A$ such that : $f(x) = \sum_{n \in A} \frac{x^n}{n!}$ is bounded

Find all subset $A \subset \mathbb{N}$ such that the function : $f : \mathbb{R}_- \to \mathbb{R}$ defined as : $$f(x) = \sum_{n \in A} \frac{x^n}{n!}$$ is bounded Note that the function $f$ is ...
0
votes
1answer
39 views

$\lim_{(x, y) \rightarrow (0,0)} \ \frac{\sin x \sin 3y}{1-\cos(x^2 + y^2)}$ [on hold]

Find the limit $$\lim_{(x, y) \rightarrow (0,0)} \ \frac{\sin x \sin 3y}{1-\cos(x^2 + y^2)}.$$
1
vote
1answer
45 views

simplifying $\sqrt{x^2}$ in a limit, when $x$ tends to $-\infty$

Sorry but I am not that good at maths, but I have one simple question. In a $\lim_{x \to -\infty}$, I want to take the $x^2$ out of a square root; after that, is it $|x^2| = x$ or $-x$? Thanks.
0
votes
1answer
19 views

Trouble with evaluating the limit of a function

We have the following function: $\mu_n(p)=\frac{1}{2}(n-1)!p^n$, where $n\geq 3$ and $0 \leq p \leq 1$. Now, we want to find lim$\mu_n(\frac{c}{n})$ as n goes to infinity. where $c$ is a constant. ...
0
votes
0answers
13 views

Equivalent definitions of convergence to 0 as $x$ goes to infinity.

Suppose for any $$\forall\epsilon>0,\exists b>a>0,\exists N,\forall x\in[a,b],\forall n\ge N: |f(nx)|\le\epsilon$$ But I do not know how this is equivalent to $$\forall\epsilon>0,\exists N,...
1
vote
2answers
83 views

$\sum_{ n = 1}^\infty \frac{1}{(2n-1)(3n-1)}$

I would like to compute the value of the following sum $$\sum_{ n = 1}^\infty \frac{1}{(2n-1)(3n-1)}$$ Clearly, it converges since $ \frac{1}{(2n-1)(3n-1)} = O(n^{-2})$. I tried to use the ...
0
votes
1answer
35 views

Simplify $\lim_{n \to \infty}\frac{(1-A)(1-B)}{(2-A-B)(A-B)}=\frac{\gamma(\gamma-1)}{2\gamma-1}?$

Let: $$A=\Gamma\left(1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}\right)$$ $$B=\Gamma\left(1+\frac{1}{1+\frac{1}{\Gamma\left(\frac{1}{n}\right)}}\right)$$ I spent a few days trying to work on the $\...
2
votes
2answers
49 views

Limit of the sum using integral

$\lim\limits_{n\rightarrow\infty}\sum_{k = 1}^{n} \frac{1}{(k+n)\sqrt{1 + n\ln({1+\frac{k}{n^2}})}}$. I can find it using integral: $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum_{k = 1}^{n} \frac{1}...
1
vote
1answer
60 views

What is the limit of $x+x^3$ as $x$ goes to 0 and is irrational? [on hold]

$ f(x)= \begin{cases} 2x&\text{if x is rational}\\ x+x^3&\text{if x is irrational} \end{cases} $ Does $\displaystyle \lim_{x \rightarrow 0}{f(x)}$ exist? If so, what is the limit? Looking ...
0
votes
0answers
31 views

Deriving the Triple Product/Cyclic Chain Rule from the limit definition of the partial derivative

I'm trying to derive the Triple Product/Cyclic Chain Rule from the limit definition of the partial derivative, and am wondering if someone could help me please? I state the Rule as follows following ...
0
votes
1answer
48 views

$\lim_{n \rightarrow \infty}\frac{x}{n}(1+\frac{x}{n})^{n-1} = ?$

I do not understand why this expression simplifies to $$xe^x$$ My intuition tells me $$\begin{align} \lim_{n \rightarrow \infty}\frac{x}{n}\left(1+\frac{x}{n}\right) ^{n-1} & = \lim_{n \...
2
votes
1answer
52 views

Adding Limits that do not exist

I'm a bit confused about the idea of adding limits when they do not exist. I'm reading a book on Calculus, and it states that $ \lim_{x \to a} \ [f(x) + g(x)] $ exists in the following cases: Both $ \...
-1
votes
2answers
56 views

How to calculate $\lim_{x\to0^-} e^{\frac{1}{x}}\left(1-\frac{1}{x}-\frac{1}{x^2}\right)$? [on hold]

I have this limit to calculate and I have some troubles with it: $$\lim_{x\to0^-} e^{\frac{1}{x}}\left(1-\frac{1}{x}-\frac{1}{x^2}\right)$$ Can you give me any hints? (I am not looking for a ...
0
votes
1answer
31 views

Calculating the limit of the sum

I would like to receive some help with the next problem: I'm trying to calculate $\lim_{n \to \infty} \sum_{k = 0}^n \frac{x^k}{k!}$, $x \in \mathbb{R}$. I know that $\lim_{n \to \infty} \frac{x^n}{n!...
1
vote
2answers
100 views

Finding $\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$

Find $$\lim_{n \to \infty} \int_0^n \frac{dx}{1+n^2\cos^2x}$$ I tried: mean value theorem. variable change with $ \tan x = t $ but I need to avoid the points which are not in the domain of $\tan$ ...
0
votes
1answer
38 views

Finding $\lim_{n \to \infty} \int_{0}^{\pi/3} \frac{1}{1+\tan^n(x)}\,dx$

Compute $$\lim_{n \to \infty} \int_{0}^{\pi/3} \frac{1}{1+\tan^n(x)}\,dx$$ I tried to do a variable change $\tan x=t$ and arrived at another integral but I haven't solved it yet.