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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 the tag “limits-colimits” instead.

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I don't know how to prove this limit

Let $x=(x_1,\cdots,x_n)$, $\alpha=(\alpha_1,\cdots,\alpha_n)$ $\sum_{i=1}^n\alpha_i=1$, we have $\alpha>0,x_i\ge 0$ for all $i$. Define $$M_t(x,\alpha)=\left(\sum_{i=1}^n\alpha_ix_i^t\right)^\frac ...
Yinuo An's user avatar
  • 356
0 votes
1 answer
28 views

Proving $\lim_{y\to 0} y^2 \ln|yx^2|=0$ using sequences

I have the function $$f(x, y) = y^2 \ln|yx^2|$$ and I want to prove that $f(x, 0)$ goes to zero but using sequences. SO I thought about this: I choose $b_n = \frac{1}{n}$, and in general any $b_n$ ...
J.N.'s user avatar
  • 65
0 votes
1 answer
30 views

Prove of rules of operation with limits that diverge

Could someone help to check my proof, I'm not sure if they are rigorous: Given ${a_n}$, ${b_n}$, and ${c_n}$ be sequences of real numbers and let k be a constant, $\lim_{n \to \infty} a_n = \infty$, $\...
DoRealAnalysis's user avatar
0 votes
2 answers
71 views

Does there exist a positive sequence with these two properties?

Let $\{x\}$ denote the fractional part of $x$. Does there exist a sequence with all positive terms $(a_n)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{(-1)^n a_n\}=1\ \ \ \text{and}\ \ \ \lim_{n\to\...
Max's user avatar
  • 926
-3 votes
0 answers
52 views

Trying to prove $(a^x)^y=a^{xy}$ using definition

In a lecture, we learned that $a^x$ is defined to be the limit of $a^{x_n}$ where $x_n$ is a series of rationals that converge to x. I was trying to prove $(a^x)^y=a^{xy}$ using solely this definition,...
PortyMart's user avatar
2 votes
2 answers
123 views

Showing that the function that returns $1$ on the rationals and $0$ elsewhere has no limit at $0$. Is there anything wrong in this proof? [duplicate]

Define $f : \mathbb R \to \mathbb R$ by $f(x) = \begin{cases}1, & \text{if } x\in \mathbb Q \\ 0, & \text{if } x \notin \mathbb Q \end{cases}$. I want to show that this function has no limit ...
Seeker's user avatar
  • 3,640
2 votes
3 answers
162 views

Evaluating $\lim_{x\to\infty}\left[\cos\frac1x\right]^{h(x)}$, where $h(x)=\frac{x^4+x^2-1}{2x+1}\sin\frac1x$

I am struggling to calculate what $h(x)$ tends to in $$\lim_{x\to\infty}\left[\cos\left(\dfrac{1}{x}\right)\right]^{h(x)}$$ where $$h(x)=\dfrac{x^4+x^2-1}{2x+1}\sin\left(\dfrac{1}{x}\right)$$ This is ...
sofischh's user avatar
3 votes
2 answers
222 views

Why can the Binomial Distribution be Approximated by a Normal Distribtuion?

As a practice problem, I am trying to prove the relationship between the Normal Distribution and the Binomial Distribution. I have seen several proofs of this before (e.g. Justifying the Normal Approx ...
konofoso's user avatar
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4 votes
1 answer
161 views

Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.

The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as $$ \log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2} $$ where $\arg(z)$ is the standard branch of the ...
Max's user avatar
  • 926
0 votes
0 answers
63 views

Landau Notation Problem

I have this function $$ K_{n} = \int_{1}^{+\infty}\frac{1}{(1+t^2)^n}dt$$ $$ \text{Let }t\geq1,t^2+1\geq1+t\Leftrightarrow\frac{1}{1+t^2}\leq\frac{1}{1+t} \text{ and for } n \in {\mathbb{N^{*}}} : \...
diplodocass's user avatar
4 votes
0 answers
54 views

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}...
EpicFaceInc100's user avatar
2 votes
3 answers
147 views

Why $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$ for polynomials $P(x)$ and $Q(x)$?

I am trying to reason why for any $2$ polynomials $P(x)$ and $Q(x)$ defined over the reals, $\lim_{x \rightarrow \infty } \frac {P(x)}{Q(e^x)} = 0$. This assertion was made in this answer which I am ...
Princess Mia's user avatar
  • 3,009
0 votes
0 answers
24 views

Integral of Poisson Kernel

This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations. The Poisson Kernel is \begin{equation} P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
Richard's user avatar
  • 89
-4 votes
0 answers
68 views

Evaluate $\lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$ [closed]

$$\displaystyle \lim_{x \to π/4} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)^2}$$ Can anyone pls help me to figure out the solution without using L'Hopital's rule as I thought if there was an alternative ...
LavN's user avatar
  • 9
0 votes
1 answer
37 views

Solving a combined limit with an $1^{\infty}$ form nested inside a 0×∞ form

I came across this limit problem: $\lim _{x \rightarrow \infty}\left\{\left(\frac{x+1}{x-1}\right)^x-e^2\right\} \cdot x^2$ Plugging this into desmos, one can see that the limit approaches $\frac{2 e^...
Afsheen's user avatar
  • 45
1 vote
0 answers
40 views

A simple clarification on convergence of functions

Definition: $\lim_\limits{\large x\ \to\ x_0 \atop \large x\ \in\ E}f(x) = L$ iff for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\vert f(x) - L \vert \leq \epsilon$ for all $x ...
Community_Digest's user avatar
-2 votes
1 answer
80 views

$\lim_{x\to\infty}\exp{(x+iy)}$ [closed]

We can write $\exp{(x+iy)}=e^x\cdot e^{iy}$. Therefor, the limit is $$\lim_{x\to\infty}{\exp{(x+iy)}}=\lim_{x\to\infty}{e^x}\cdot e^{iy}=\infty\cdot e^{iy}$$ My question is: what is this and what its ...
Dionysis Balasis's user avatar
1 vote
2 answers
59 views

Tiling potential of the primitive triplets

What is the maximum percentage of an infinite plane that could be tiled with primitive Pythagorean triangles, if every triangle may be used at most once? My intuitive guess is that it can’t reach 100%,...
Nalacram's user avatar
1 vote
1 answer
67 views

Problem E1 in Engels's "Problem Solving Strategies" [duplicate]

I am trying to solve problem E1 in Engels's book on the invariance principle. He includes a write-up of a solution, but it seems to me to be missing details. I'll paraphrase what he wrote. The setup ...
Cardinality's user avatar
  • 1,243
-2 votes
1 answer
61 views

If the integral of a monotonic f converges, does it mean f approaches 0? [closed]

I have come across this question: Say $f(x): [0,\infty) \rightarrow \mathbb{R}$ is monotonic non-increasing, and $\int_{0}^{\infty} f(x)dx$ converges. Does it mean that $\lim_{x\to\infty}{f(x)}=0$? If ...
Nadav Menirav's user avatar
3 votes
1 answer
42 views

Why do we need the right-continuity of $F$ at $a$ to prove $V_F[a,c]=\lim_{b\to a^+}V_F[b,c]$ where $F$ is of finite variation?

This is a continuation of this post. The book claims the following without proof: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. Suppose also that $a<c$ and $F$...
Shenron's user avatar
  • 75
2 votes
3 answers
56 views

Proof of $V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,b]$ where $F$ is of finite variation.

I am reading the proof of the following result: Proposition$\quad$ Suppose that $F:\mathbb{R}\to\mathbb{R}$ is of finite variation. If $b\in\mathbb{R}$, then $$ V_F(-\infty,b]=\lim_{a\to-\infty}V_F[a,...
Shenron's user avatar
  • 75
1 vote
1 answer
32 views

Vector algebra and limits question

The following question just occurred to me yesterday: The zero or null vector, $\boldsymbol{\vec 0}$, has neither magnitude nor direction, but for a unit vector $\boldsymbol{\vec r}$ and some scalar ...
questing-monkey's user avatar
2 votes
2 answers
209 views

Is $\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$ not convergent?

I need to calculate the limit (if it exist) $$\lim_{n\to\infty}\left\{n!\sum_{k=1}^{n!}\frac{1}{k^{\frac{3}{2}}}\right\}$$ where $\{x\}$ denotes the fractional part of $x$, $n!$ is the factorial of $n$...
Max's user avatar
  • 926
1 vote
2 answers
68 views

How to prove that a strictly increasing sequence of whole numbers approaches infinity? [closed]

Prove that if $(a_n)$ is a strictly increasing sequence ($a_{n+1}> a_n$ for all $n \in \mathbb{N}$), and $a_n \in \mathbb{Z}$ for all $n \in \mathbb{N}$, then $\lim\limits_{n \to \infty} a_n = \...
Magnivul's user avatar
-4 votes
1 answer
268 views
+50

Like Ramanujan : Strange limit and a remarkable behavior .

Problem: Let $f_a\left(x\right)=a^{3-\sqrt{1+2x!\sqrt{1+3x!!\sqrt{1+4x!!!\sqrt{\cdot\cdot\cdot\sqrt{1+(k-1)x!\cdots!\sqrt{1+kx!!\cdots!}}}}}}}, g_a(x)=\left(\frac{f_a\left(x\right)}{f_a\left(0\right)}\...
Ranger-of-trente-deux-glands's user avatar
0 votes
1 answer
76 views

How to solve this limit $\lim_{n\to \infty } \sum_{k=0}^n 1/[(2k+1)(2k+3)]$ [closed]

Help please: $$ \mbox{I tried to do it like this}\quad \lim_{n \to \infty}\sum_{k = 0}^{n}{1 \over \left(2k + 1\right)\left(2k + 3\right)}$$ but I don't know how to continue.
intenziven's user avatar
-1 votes
1 answer
65 views

Solution verification of $\lim_{n \to \infty} {x}^{\left( {n}^{x / n} \right)}=x$. [closed]

Let $x\in\mathbb{R^{+}} $ and $n\in\mathbb{N} $. Consider the function $$f(x)=x^{\left(n^{x / n} \right) }.$$ For $x=0$, $$\lim_{n\to\infty} f(x) =\lim_{n\to\infty} 0^{(n^{0})}=0.$$ For $x≠0$ and ...
Bacha Fethi's user avatar
1 vote
1 answer
38 views

Evaluating a one-sided limit that goes to negative infinity, where the denominator goes to $0$

I attached my handwriting explaining the issue, hope that's okay. Basically, I can "intuitively" find the limit and explain it on paper. However, I'm not sure if this "approach" is ...
Aviv Cohn's user avatar
  • 469
5 votes
0 answers
166 views

Prove or disprove the limit of a sequence is negative.

I have a sequence of positive numbers $\{f_k\}$ such that all the odd terms sum up to 1 and so do all the even terms, i.e. $\sum_{k=1}^{\infty}f_{2k-1}=\sum_{k=1}^{\infty}f_{2k}=1$, and $1>\sum_{i=...
Jake ZHANG Shiyu's user avatar
0 votes
0 answers
41 views

Leibniz rule when integrand has discontinuity

For $0<y<1$ consider, $$F(y)=\int_0^y \frac{-\log^3 x}{1-x} dx$$ I need to prove that $$F'(y)=\frac{-\log^3 y}{1-y}, 0<y<1$$ By definition $$F'(y)=\lim_{h\to 0}\frac{F(y+h)-F(y)}{h}$$ So ...
Max's user avatar
  • 926
-1 votes
0 answers
34 views

Evaluate the limit without l hopital [closed]

$\displaystyle \lim_{x \to 1} \frac{\sqrt{2}- \cos(x)- \sin(x)}{(4x-π)}$ Please provide the solution to the question without using L hospital.
LavN's user avatar
  • 9
0 votes
1 answer
58 views

Compute $\lim_{n\to\infty}(\frac{1+h^{\frac{1}{n}}}{2})^{n}$ [duplicate]

Compute $$L=\lim_{n\to\infty}\left(\frac{1+h^{\frac{1}{n}}}{2}\right)^{n}$$ where $h>0$. With a computer to compute it numerically, $L$ seems to tend to $\sqrt{h}$, but I could not figure it out ...
Shine's user avatar
  • 143
2 votes
3 answers
86 views

Why does $\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)} = L \implies f = \Theta(g)$ not hold when $L=0$?

I am currently seeing a contradiction from my use of the "theorem" For any $2$ functions $f : \mathbb{Z}^{+} \rightarrow \mathbb{R}^{+}$ and $g: \mathbb{Z}^{+} \rightarrow \mathbb{R}^{+}$, ...
Princess Mia's user avatar
  • 3,009
1 vote
0 answers
46 views

Comparing two sets : if $u$ is in the set, so is $2u +1$ vs $2u + 5$ (extended mersenne numbers followup)

Consider these two sets of odd positive integers. SET 1 : constructed by these rules : a) $1$ is in the set. b) if $x$ is in the set, then so is $2 x + 1$. c) if $x$ and $y$ are in the set then so is $...
mick's user avatar
  • 16.4k
1 vote
1 answer
93 views

Does there exist a sequence such that $\lim_{n\to\infty} \{(-1)^na_n^2\}=1$, $\{x\}$ is the fractional part of $x$

Let $\{x\}$ denote the fractional part of $x$. I need an example of a sequence with all positive terms $(a_n^2)_{n\geq 1}$ such that $$\lim_{n\to\infty} \{(-1)^na_n^2\}=1\ \ \ \text{and}\ \ \ \lim_{n\...
Max's user avatar
  • 926
0 votes
0 answers
27 views

Metric entropy for the parametric family $1-e^{-\theta x}$

Consider the parametric class of function $$ f_\theta (x) = 1-e^{-\theta x} $$ defined on the interval $[0, 1]$, and $\theta \in [0,1]$. Using the sup-norm, we can bound the $\delta$-covering number $...
Mondayisgood's user avatar
2 votes
1 answer
96 views

The area of a inscribed polygon tends to the area of the circle

As it is broadly known, given a circle of radius $r$, its area is equal to $\pi \cdot r^2$. My goal is to prove this formula using inscribed polygons. Let´s call $n$ the number of sides of a regular ...
IkerUCM's user avatar
  • 402
1 vote
4 answers
162 views

Calculating: $\lim_{n\to\infty}{\int_{3}^{4}}( \sqrt{-x^2+6x-8})^{n}dx$ [duplicate]

Calculate: $\lim_{n\to\infty}\int_{3}^{4} \left(\sqrt{-x^2+6x-8}\right)^{n}{\rm d}x.$ I've tried to change the variable and I took it as $y=x-3$. I have changed the limits of integrations, for $x=3$, ...
Emil Cohen's user avatar
1 vote
1 answer
102 views

Do you need L'Hôpital's rule to prove Taylor's formula?

I recently read a Quora answer. The answerer was asked to solve the limit $$\lim_{x\to0}\frac{\cos x-e^x}{\sin x}$$ without using L'Hôpital's rule. The answerer used the Taylor series expansion of the ...
Elvis's user avatar
  • 610
1 vote
0 answers
48 views

Confusion about $\lim\sup$ and its definition as the greatest limit point

I posted a question a few days ago and the most voted answer uses $\lim \sup$, a concept I was not familiar with. I decided to jump ahead and read about $\lim\sup$ to understand the answer, but one ...
ten_to_tenth's user avatar
  • 1,426
7 votes
2 answers
120 views

Prove $\lim_{c\to\infty} P(X-\varepsilon<Y<X+\varepsilon\mid X>c,Y>c)=1$ for all $\varepsilon>0$ if $X,Y$ are i.i.d. Normal$(0,1)$?

Suppose random variables $X$ and $Y$ are i.i.d. Normal$(0,1)$. Consider the following events, where $\varepsilon>0, c>0$: $$\begin{align*} Q&=\{(x,y)\in\Bbb R^2: x>c, y>c\}\\ C&=\{(...
r.e.s.'s user avatar
  • 15k
-3 votes
0 answers
37 views

Finding a weaker speed of divergance for a real sequance [closed]

Let \begin{equation*} \lim_{n} \frac{1}{n} \sum_{k=1}^{n-1} \frac{\ell_{k-1}}{\ell_{k}} \leq \gamma <1,~ and ~~ \frac{\ell_{k-1}}{\ell_{k}} \leq \frac{3}{2} ~ \forall ~ k \in \mathbb{N} \end{...
Rizwan Ullah's user avatar
6 votes
1 answer
125 views

Is there a closed form for the quadratic Euler Mascheroni Constant?

Short Version: I am interested in computing (as a closed form) the limit if it does exist: $$ \lim_{k \rightarrow \infty} \left[\sum_{a^2+b^2 \le k^2; (a,b) \ne 0} \frac{1}{a^2+b^2} - 2\pi\ln(k) \...
Sidharth Ghoshal's user avatar
7 votes
3 answers
418 views

Showing that $f(x)=\lim_{n\to{\infty}}(\frac{x}{n}+1)^n$ has the property $f(a+b)=f(a)\cdot f(b)$

On the way to show that the function $f(x)=\lim_{n\to{\infty}}\left(\frac{x}{n}+1\right)^n$ has this property: $f(a+b)=f(a)\cdot f(b)$, a math professor explained me that: $\lim_{n\to{\infty}}\left(\...
lazare's user avatar
  • 277
0 votes
2 answers
198 views

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 $\...
Aryaan's user avatar
  • 281
2 votes
4 answers
111 views

Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule?

Is it possible to show that for $q>0$, $\lim\limits_{x\to\infty}\dfrac{(\ln{x)^p}}{x^q} = 0$ without using L'Hopital's Rule? Applying L'Hopital's Rule repeatedly until the numerator becomes a ...
ten_to_tenth's user avatar
  • 1,426
0 votes
0 answers
115 views

Prove that this limit is equal to $\sqrt{2}$ for the function $f(x)=x^2-2$ for an arbitrary seed point $s$.

Mathematica knows that: $$ s + \frac{1}{1-\lim_\limits{n\ \to\ \infty}\left[\frac{\displaystyle\sum _{k=1}^n \frac{(-1)^{k-1} \binom{n-1}{k-1}}{f\left(k/n + s -1/n\right)}}{\displaystyle\sum _{k=1}^...
Mats Granvik's user avatar
  • 7,420
-1 votes
0 answers
48 views

Limit of square root, not working as expected [duplicate]

Hey i have this function, and I don't understand why I get wrong limit if I insert x into the square root, even though it's correct algebraic to insert it. $$ \frac{\sqrt{x^2 + 9}}{x} $$ The first ...
miiky123's user avatar
  • 215
0 votes
3 answers
96 views

$\lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=1}^{n} \frac{k}{\ln(k+1)}$ [duplicate]

I am interested in the limit $\lim_{n \rightarrow \infty} \frac{1}{n^2} \sum_{k=1}^{n} \frac{k}{\ln(k+1)}$. While I suspect the limit to be 0, I cannot prove it rigorously. By intuition we have $\sum_{...
Tom Lucas's user avatar

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