Questions on the evaluation and properties of limits.

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1
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0answers
17 views

under which conditions this equality holds

Consider $f : [0,\infty) \rightarrow \mathbb{R}$ be a function such that $\lim_{t\rightarrow \infty} f(t) = 0$. I was wondering if the following relation holds $$lim_{t\rightarrow\infty}\int_0^t ...
2
votes
2answers
22 views

How to prove this sequence converges

Here is a problem in analysis: Suppose $x_n\geq0$ and for all $n$, there is $$ x_{n+1}\leq x_n+\dfrac1{n^2} $$ Prove that $x_n$ converges. My approach: it is easy to prove $x_m-x_n\leq ...
0
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0answers
10 views

Find lower bounds and upper bounds in a time series

Given the following dataset, I want to group based on the frequency (more like bell curve), and find lower bound and a upper bound. ...
1
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1answer
28 views

Wrong derivation of limit of Cesàro mean

It's known that $$\lim_{n\rightarrow\infty}x_{n}=a\Rightarrow\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}x_{i}}{n}=a$$ Consider the following derivation: ...
0
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2answers
27 views

Calcultaing function limit of a limit sequence

How do I even start? $$\lim_{x \to 1^+} \lim_{n \to \infty}{x^n \over x^n + 7}$$ I see that it should be $1$ but how do i prove it?
3
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2answers
48 views

How to prove $\lim \limits_{n\to\infty}(n+1)\int_{0}^{1}x^nf(x)dx=f(1)$

I need help to prove this in real analysis. I think it uses IMVT, but not sure how to do it. Let $f(x)$ be a real valued continuous function on $[0,1]$. Show that $$ \lim ...
0
votes
2answers
20 views

Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?
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1answer
20 views

Calculate the limit of the sequence by applying the limit laws?

I'm not sure how to approach this problem since its a bit different to the usual questions about calculating limits .
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4answers
34 views

Use the definition of a limit to prove that the limit is equal to zero?

All I can think of to start is to state that: $$|n-∞| < \delta \Rightarrow |(c/n^2)-0| < \epsilon$$ But I don't know where to go from there
0
votes
3answers
65 views

difficult limit with a improper integral

It is assigned at my calculus class the following problem. problem: Evaluate the following limit $$\displaystyle \lim_{n \to \infty} \int \limits_{\frac{1}{(n+1)^2}}^{\frac{1}{n^2}} ...
0
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0answers
10 views

Equivalence between $\limsup\frac{f(x)}{g(x)^{2-\epsilon}}=0$ and $\liminf \frac{|\log(f(x))|}{|\log(g(x)^2)|}\geq 1$

Suppose that $f,g\geq 0$ are positive functions on $(0,\infty)$, and assume that $g(x)\rightarrow 0$ as $x\rightarrow\infty$. I am trying to prove that the following two claims are equivalent. I have ...
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2answers
70 views

Limits and Trigonometry

Consider an function $f$ , defined as : $$f^k (\theta) =\sum_{r=1}^n \left( \frac{\tan \left( \frac {\theta}{2^r} \right) }{2^r} \right)^k +\frac 1 3 \sum _{r=1}^n \left( \frac { \tan \left( ...
0
votes
0answers
21 views

Limit of Distribution, Hilbert Transform

I want to peform a distributional limit of the following distribution: $\frac{2 i}{x^2 \epsilon} e^{-(t + x)^2/(4 \epsilon)} (F(\frac{t - x}{2 \sqrt{\epsilon}}) - F(\frac{t + x}{2 ...
1
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1answer
21 views

Understanding part of a proof of l'Hôpital's Rule

I am studying a proof of a simple l'Hôpital's rule. The theorem is as follows: Let $x_0 \in (a, b).$ Suppose that $f, g$ are differentiable on $(a, b)$. Suppose $\lim_{x\to x_0}f(x) = \lim_{x\to ...
0
votes
2answers
31 views

Limit of a sequence defined by a non-linear recurrence relation

How can one find the limit for the sequence $\{x_n\}^{+\infty}_{n=0}$ where $$x_0 = 0, x_1 = 1, x_{n+1} = \dfrac{x_n + nx_{n-1}}{n+1}$$ By computing the values I came to the conclusion that it ...
3
votes
4answers
528 views

Prove that limit doesn’t exist anywhere? [on hold]

I'm doing some practice problems and am having trouble answering these problems: Consider the following function $$f(x)=\begin{cases}1, & \text{if } x\in \Bbb Q\\ -1, & \text{if } x\in \Bbb ...
6
votes
2answers
67 views

A simple way to find $\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$

I was reading an exam paper used to identify gifted high-school students, and I encountered the following problem: $$\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$$ Using ...
0
votes
1answer
41 views

Help with limit of function

How can I calculate the limit $$\lim_{x \to \infty} x^{3/2}( \sqrt{x+1}+ \sqrt{x-1}-2 \sqrt{x})$$ I had ideas like using Laurent series, but I dont think I am allowed since its an elementary course, ...
1
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1answer
57 views

How to find $\frac{0}{0}$ limit without L'Hôpital's rule

I am having trouble solving this limit. I tried applying L'Hôpital's rule but I got $\frac{0}{0}$. $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + ...
0
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2answers
56 views

Prove $\lim_{x\rightarrow1} \frac{x}{x+1}=\frac{1}{2}$ using $\epsilon$-$\delta$ [on hold]

Prove $\lim_{x\rightarrow1} \frac{x}{x+1}=\frac{1}{2}$ using $\epsilon$-$\delta$ I am thinking I should choose $\delta = \min (1, 2\varepsilon)$ but I might be way off.
0
votes
3answers
47 views

Simple calculus question (limits)

So I have to calculate the following limit $$\lim_{u\downarrow 1}\frac{\frac{2u}{3}-\frac{2}{3u^2}}{2\sqrt{\frac{u^2}{3}-1+\frac{2}{3u}}}.$$ I tried to use L'Hopitals rule, but it doesnt work it ...
1
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2answers
51 views

Finding the limit of a recursively defined sequence (recurrence relation). Specifically and generally

Whilst reading Goldrei's Classic Set Theory, I have come across a recursively defined sequence $a_0=0, a_1=1, a_n=\frac{1}{2}\left(a_{n-1} + a_{n-2} \right) $ The first few terms of which are: $0, ...
0
votes
4answers
42 views

Proving a recursive sequence is bounded

I'm proving that the limit of the following recursive sequence is $\dfrac{10}{9}$: $$s_0=1,\,s_n=s_{n-1}+\frac{1}{10^n}\quad\text{for }n\ge1$$ Showing that the sequence is monotonic was easy enough, ...
1
vote
4answers
36 views

Limit of a rational function with radicals [on hold]

How do I solve this limit: $$\lim_{x\to0}\frac{\sqrt{x^2+p^2}-p}{\sqrt{x^2+q^2}-q}$$
0
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2answers
32 views

How do we show that limit $\frac{x^6-x^2sin(\frac{1}{x^2})}{x^4}$ as x tends to $0$ does not exist?

How do we show that the limit of $\frac{x^6-x^2sin(\frac{1}{x^2})}{x^4}$ as x tends to $0$ does not exist? I thought maybe we should consider two sequences that tend to 0 and show than $f(a_{n})$ and ...
2
votes
1answer
43 views

If $f$ satisfies certain conditions, then show that $\lim_{x \rightarrow \infty}{\frac{f(x)}{x}}=0$

Suppose $a\in \mathbb{R}$, $a \in (0,1)$ and a function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying the following property: $(1)$ $\lim_{x \rightarrow \infty}{f(x)}=0$ $(2) \lim_{x \rightarrow ...
8
votes
1answer
90 views

Limits, Taylor expansion

Find the limit: $$ \lim_{x\to\infty} \frac{\displaystyle \sum\limits_{i = 0}^\infty \frac{x^{in}}{(in)!}}{\displaystyle\sum\limits_{j = 0}^\infty \frac{x^{jm}}{(jm)!}} $$ for $n$, $m$ natural ...
1
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3answers
52 views

How to solve this particular indetermination: $0*\infty$

The limit in question is: $$\lim_{x\to\infty} 2n\left(\sqrt{n^6+5n^2}-n^3\right)$$ By looking it up on wolfram alpha I found out the answer is 5 but I am not so sure how to arrive to it. I tried to ...
0
votes
2answers
70 views

Evaluating $\lim_{x \to\infty }\left (x^{f(x)}-x \right )$

I have a question about evaluating the limit: $$\lim_{x \to\infty }\left(x^{f(x)}-x \right)$$ where: $f(x)$ is a continuous map from the positive reals to the positive reals , and ...
0
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1answer
40 views

Real Analysis Question- Differentiability on an interval

So this is the question I am trying to answer... At $(-2,0)$ and $(0,2)$ we just differentiate using normal rules of calculus, yes? Here is my attempt for at $x=0$. Is this correct? For d)I think ...
2
votes
2answers
57 views

For which $x\in\mathbb{R}$ is the series of general term $a_n = x^{n!}$ convergent?

I firstly found the simplified form of $\frac{a_{n+1}}{a_n} = |x|\cdot|x^n|$ and used this to establish the end points $-1\lt x\lt 1$. I then tested the end points by finding the limit to infinity of ...
1
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1answer
27 views

Find derivative using the definition of the derivative as a limit

$$f(x) = \frac{1} {\sqrt{x}}$$ find $f'(x)$ using the definition of the derivative as a limit. I know that $$ f'(x) = \frac{(x + \delta)^{-1/2} - (x)^{-1/2}}{\delta} $$ as $\delta$ goes to $0$. ...
1
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0answers
46 views

How to Find the pointwise limit of $(f_n)$

For $x \in [0, \pi/2]$, if $$f_n(x) = \frac {nx} {1+n\sin(x)}$$ how do you find the pointwise limit of $(f_n)$ ?
1
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0answers
19 views

Limit of correlation function using transfer-matrix method

This question is about a stochastic process theory. I really very bad in this topic. That's why I have to ask for help. I may mistranslate some terms but I'll do my best to give you right information. ...
0
votes
0answers
55 views

Finding the limit of this specific series

So, I have to calculate: $$\lim _{ n\to\infty } \prod_{k=2}^{n} \Big(2-\sqrt[k]{2}\Big)$$ So far I managed to get to: $$\lim _{ n\to\infty } \sum_{k=2}^{n}\Big(1-\sqrt[k]{2}\Big)$$ Any help will ...
1
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2answers
67 views

How to find this limit $\lim\limits_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$ [on hold]

How would I find this limit? $$\lim_{(x,y) \to (1,1)} \frac{y-x^4}{y^3-x^4}$$
0
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2answers
39 views

Computing the following limit

I am required to compute this result in order to figure out another result. So I'm trying to show that $$\lim_{n \rightarrow \infty}(1-e^{-(\ln(n)+x)})^n = e^{-e^{-x}}$$ I'm hestiant to use the ...
0
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0answers
21 views

It is correct this definition of limit of a function?

I have a definition of the limit of a function in some point $\alpha$ for metric spaces on this manner: We have two metric spaces $(E,d)$ and $(F,p)$; $A\subset E$, $f:A\to F$. Then ...
3
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1answer
23 views

How to calculate the continuum limit of a discrete system?

The question is based on the following excerpt from the book "Symmetries and Integrability of Difference Equations" Link: Book Excerpt Consider the discrete equation ...
4
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7answers
89 views

Calculate $\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$

Question: Calculate $$\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$$ using substitution, cancellation, factoring etc. and common standard limits (i.e. not by L'Hôpital's rule). Attempted ...
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3answers
47 views

$\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ [on hold]

What would be the best way to find $\lim _{ x->\infty }{ [(x+2)\arctan(x+2) } -(x)\arctan(x)]$ ?
2
votes
3answers
146 views

Determining the limit of this series

$$ \sum_{n=0}^\infty \frac{(-2)^n + 2^{3n}}{3^n4^n} = $$ $$ \sum_{n=0}^\infty \frac{(-1)^n2^n}{3^n4^n} + \sum_{n=0}^\infty \left(\frac{2}{3}\right)^n = $$ $$ \sum_{n=0}^\infty (-1)^n\frac{1}{6^n} + ...
3
votes
1answer
48 views

length of the curve $y=x^n$ in the unit square

Let $l_n$ be the length of the curve $y=x^n$ in $[0,1]\times[0,1]$. Then obviously $\lim_{n\to\infty}l_n = 2$. What about $\lim_{n\to\infty}(n(2-l_n))$ ? The formula $l_n = ...
1
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3answers
147 views

Limit of a Sine Function

Calculate via the limit definition: $$\lim\limits_{x \to \frac {\pi}2^-} \frac{\sin^2(\frac {\pi}2-x)}{\sqrt{\pi-2x}},$$ I tried to calculate this limit using the definition of a limit and got ...
1
vote
4answers
78 views

Calculate the limit as $x\to0$

I need to calculate the limits as $x$ tends to $0$ For the first one, I get that the limit is zero, by splitting it up into $x^3(\sin(1/x))$ and $x^3(\sin^2(x))$ and using the sandwich theorem on ...
0
votes
1answer
21 views

Determine whether the limit exists and justify answer

Usually for these types of questions, I use sequences of functions to show that the limit does not exist, but I don't think I can do this here? I feel like the limit should be zero, but I don't know ...
5
votes
1answer
66 views

A tough limit problem involving $1/(\sin x - \sin a)$ and its generalization

Long back I had encountered the following problem in Hardy's Pure Mathematics (originally from the infamous Mathematical Tripos 1896): If $$f(x) = \frac{1}{\sin x - \sin a} - \frac{1}{(x - a)\cos ...
0
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1answer
80 views

How to compute $\lim_{n\rightarrow\infty}e^{-n}\left(1+n+\frac{n^2}{2!}\cdots+\frac{n^n}{n!}\right)$ [duplicate]

There is a probabilistic method to solve it. But I am not familiar with probability. I am trying to compute it by analytic method, such as using L Hospital's rule or Stolz formula, but they are not ...
4
votes
5answers
96 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
1
vote
1answer
36 views

Prove that $f$ in monotonic

In my assignment I have to prove the following: Let $f$ a continuous function in $\Bbb R$. Prove the following: if $|f|$ is monotonic increasing, in R then $f$ is monotonic in R. ...