Questions on the evaluation and properties of limits.

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Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
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1answer
31 views

Problem with a limit with a integral in it

Suppose that the temperature in a long thin rod placed along the $x$-axis is initially $\frac{C}{2a}$ if $|x| \leq a$ and $0$ if $|x| > a$. It can be shown that if the heat diffusivity of the rod ...
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1answer
10 views

Does convergence of bounded, increasing sequences generalize from the set of real numbers to arbitrary partially ordered sets?

If $t$ is any real number, then I can find a strictly increasing sequence $(t_n)$ of real numbers converging to $t$. E.g. $t_n = t - \frac{1}{n}$ would do. Does this generalize to arbitrary partially ...
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3answers
44 views

Finding the limit to infinity of a summation

The following problem was featured as a challenge in a previous exam paper and has left me stumped. Compute the following limit: $$ \lim_{n \to \infty} \frac{1}{n^{2013}} \sum_{k=1}^n k^{2012} $$ ...
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1answer
11 views

Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be ...
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2answers
35 views

Limit of indeteminate form $1^{(∞)}$

If we consider the function $f(x)=[(ax+1)/(bx+2)]^{x}$ where $a$,$b$ >$0$ and a I tried as follows]1 But at end i got stuck .
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4answers
35 views

Finding the limits of a trig function

I have been struggling with finding the following limit: $$ \lim_{x\to \pi} \frac{\cos x + 1}{x - \pi} $$ Use of L'Hospital's rule is not permitted. Thanks
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1answer
25 views

Show that if $|s_{mn}-S|<\varepsilon$ then $|\lim_{n\to\infty}s_{mn}-S|\le\varepsilon$

This is the exercise 2.8.5 of the book Understanding analysis of Abbott. To put you in context I have that The sequence of partial sums $(s_{mn})$ is absolutely convergent, and the definition is ...
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1answer
32 views

Show that sum is convergent [on hold]

Assume that $\sum\limits_{i=1}^{\infty} a_i = L$ . Show that $\lim_{x\to\infty} (S_1+S_2+...+S_x)/x = L$. Where $S_n=\sum\limits_{i=1}^{n} a_i$
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1answer
44 views

Limit of a Riemann sum: $\lim_{n\to\infty} {n^5 \sum^n_{r=0}\frac1{(n^2+r^2)^3}} $

Required to find $\lim_{n\to{\infty}} {n^5 \sum^n_{r=0}\frac{1}{(n^2+r^2)^3}} $ $\lim_{n\to{\infty}} \frac{1}{n} \sum^n_{r=0}(\frac{n^2}{n^2+r^2})^3$ $\lim_{n\to{\infty}} \frac{1}{n} ...
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2answers
34 views

Finding limit with improper integral

How should I approach this question? $$\lim_{x\to0}\frac{1}{x}\int_1^{1+x}\frac{\cos t}{t} \, dt$$ I tried to use L'hospital and that gave me $-\sin(0) = 0$ The correct answer is $\cos 1$. Did I ...
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1answer
27 views

Finding Limit Points?

I have two sets: $ A = ({{ (-1)^n + 2/n : n = 1, 2, 3, ...}}) $ and $ B = ( x \in \mathbb{Q} : 0 < x < 1 ) $ How does one go about finding the limit points for these sets? Would I just do $ ...
3
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0answers
38 views

About a curious nested radical representation for $\cos 1^\circ$

I have found the following nested radical representation. By using the triple angle formula for the cosine, $\cos 3\theta$, and making $\theta = 1^\circ$, we get the cubic equation $ 4x^3-3x = \cos ...
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0answers
20 views

A complex sequence of functions $(f_n)$ is continuously convergent iff it's compactly convergent against a continuous function

Let $G \subseteq \mathbb{C}$ be a region in $\mathbb{C}$, i.e. $G$ is open, nonempty and connected, and let $f_n: G \to \mathbb{C}$ be a sequence of complex-valued functions, with $n \in \mathbb{N}$. ...
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1answer
41 views

If $f\colon\mathbb{R}\to\mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2$ for every $x\in\mathbb{R}$, then $f$ is differentiable at 0.

If $ f\colon \mathbb{R} \to \mathbb{R}$ satisfies $\lvert f(x)\rvert\le x^2 $ for every $x \in \mathbb{R} $, then $f$ is differentiable at $0$. The solution provided uses delta-epsilon to prove ...
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5answers
63 views

Find the limit $\lim_{x \to 1} \left(\frac{p}{1-x^p} - \frac{q}{1-x^q}\right) $ $p ,q >0$

I Know series expansion and L'Hospital's rule . But here both of them are not of any help.
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2answers
48 views

Using L'hopital's rule to prove differentiability

Question: Define $$ f(x) = \left\{\begin{aligned} & \frac{2x\cos(x)}{x+\sin(x)} &&: x \ne 0 \\ &1 &&: x = 0 \end{aligned} \right.$$ Show $f'(0)$ exist and find the ...
2
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1answer
46 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) ...
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1answer
44 views

Is this estimate true or not true?

Let $\varepsilon>0$. Let $\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ the standard normal density function. Then $$\lim_{\varepsilon\to 0}\int_0^1 \frac{1}{\sqrt{x}}\left[ ...
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1answer
57 views

The range of a twisted composition of sines

Let $f(x) = \sin \left( \frac \pi 6 \sin \left( \frac \pi 2\sin x \right) \right)$ and $g(x) = \frac \pi 2 \sin x$ for all $x \in \Bbb R$. Which of the following are true? The range of $f$ is ...
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2answers
27 views

How to find limit of the sequence

A sequence is defined by recurrence relation $f_{n+1}=\frac{3}{7}f_n+8$ with $\mu_0=-14$, then what is the limit of the sequence? $14$ $-14$ $\frac{-3}{7}$ $\frac{3}{7}$ My attempt: As wiki ...
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1answer
60 views

TIFR GS 2015 computer science: $G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$

Following expression was asked to be evaluated in TIFR GS 2015 exam, $$G = \lim_{n\to\infty}(n+1)\int_{0}^{1} x^{n} f(x) dx$$ where $x \in [0, 1]$ and $f(x)$ be any real valued continuous function. ...
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3answers
932 views

What am I doing wrong in calculating the following limit?

$$\lim_{x\to-2} \frac{x+2}{\sqrt{6+x}-2}=\lim_{x\to-2} \frac{1+2/x}{\sqrt{(6/x^2)+(1/x)}-2/x^2}$$ Dividing numerator and denominator by $x \neq0$ ...
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1answer
40 views

If $g(x) = xf(x^2)$ and $f(x)=\sum_{n=0}^\infty \sin(\frac{\pi}{n+2})x^n$, what is $f^{(20)}(0)$ and $g^{(35)}(0)$?

My task is this: (i)Let $$f(x)=\sum_{n=0}^\infty \sin\left(\frac{\pi}{n+2}\right)x^n.$$ Find $f^{(20)}(0)$ and $g^{(35)}(0)$ when $g(x) = xf(x^2)$. (ii)Find ...
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2answers
25 views

create function from graph/from limits

I am a Calculus I student and we are into our second week and finishing up limits. I know how to create a graph from limits, and I know that for example a parabola would match with a quadratic ...
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38 views

asymptotic behavior of functions which are defined recursively

For $x\in[0,1]$, define $f_1(x)=1$ and $$ f_{n+1}(x)=(1-x)^n\left(nx+1\right)+f_n(x)\left(1-(1-x)^{n+1}\right)\;\;\mbox{for}\;\;n=1,2,\dots. $$ I want to study some asymptotic results of $f_n$. For ...
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1answer
24 views

Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$

I have the following exercise: Let $f:A\to N$, show that if there exists $\lim_{x\to a}f(x)$ we have $b\in \overline{f(A)}$ I don't know what $b$ is meant to be, there's a typo in this exercise. I ...
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2answers
18 views

Using the definition of the convergence of the sequence, how can I prove that this sequence converges to the limit?

I suppose what I am confused most on here is the algebraic method. $\lim_{x\to\infty} $ $ 2n^2/(n^3 + 3) $ = 0 I have set it up so far: Let $ \epsilon > 0 $ be given. I set up the equation ...
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1answer
21 views

If $f$ is continuous and moderate decreasing, then Fourier transform of $f$ is continuous.

If $f$ is continuous and of moderate decrease, show that $\hat{f}$ is continuous. My attempt: $$ \hat{f}(\omega+h)-\hat{f}(\omega) = \int_{-\infty}^\infty f(x)e^{-2\pi ix\omega}(e^{-2\pi ix h} - ...
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1answer
20 views

$A$ is an open subset of $M$ $\iff$ ($x_n\to a\implies x_n\in A$ for large $n$)

My definition of an open subset $A$ of $M$ is the one that for every $x\in A$, there is an open ball contained in $A$. Now, suppose that $x_n\to a$. By definition, $\forall \epsilon>0$ there exists ...
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2answers
27 views

$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$

I need to prove the following: $$0\le d(x_n, a)<\frac{1}{n}\implies \lim x_n = a$$ It looks pretty intuitive since I can make $\frac{1}{n}$ as small as I want, thusk making $a$ as close as to ...
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0answers
23 views

Questions about proof of $\lim x_n = a, \lim y_n = b\implies \lim x_n+y_n = a+b$ in a normed vector space

I need to prove that, in a normed vector space $E$, we have: $$\lim x_n = a, \lim y_n = b\implies \lim (x_n+y_n) = a+b$$ and: $$\lim\lambda_n = \lambda, \lim x_n = a \implies \lim \lambda_n\cdot ...
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1answer
20 views

Calculator limits on a parabola

Hi guys I'm making Patrick Star for a graphing project. Anyways I'm using a parabola for his head on my TI-84 but when I set limits on it, it graphs a straight line. So the equation itself is ...
1
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1answer
20 views

When does $\max \lim{a, |b|} \leq a + \max \lim {|b|}$?

Let $\alpha_\delta >0$ be a quantity that depends only on $\delta$ and let $I_\varepsilon$ be defined as follows: $$I_\varepsilon = \alpha_\delta + \beta_{\varepsilon, \delta}$$ where ...
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3answers
44 views

Is it allowed to “ignore” $\lim$ in this case?

I have to prove $\lim_{x\to0}(1+x)^{1/x}=e$. Now I would like to perform the following operations: $$\lim_{x\to0}(1+x)^{1/x}=e$$$$\lim_{x\to0}\ln(1+x)^{1/x}=1$$$$\lim_{x\to0}\frac{\ln(1+x)}x=1$$ ...
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2answers
82 views

Am I using sandwich theorem incorrectly?

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit ...
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0answers
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PLZ HELP :) : CALCULER PLZ HELP [on hold]

\lim _{n\to :+\infty :}:\sum _{k=1}^n:\frac{k}{k^2+n^2}
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2answers
48 views

$\ln(x)$ and Big O notation

I have tried to assert that $\ln(x)=O(x^0)$ a few times, but it seems fairly obvious that this statement should be false, and so I've been faced with some rightful speculation. My reason is that ...
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2answers
72 views

Prove for a close subset of $\mathbb R$

Let $S=\{x\in \mathbb{R}\mid 0\leq x\}$. Prove that $S$ is a closed subset of $\mathbb{R}$. I know I need to show that $\forall x \exists x_n \xrightarrow[n \to \infty]{} x\in S$, but I have no ...
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4answers
110 views

Extended $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ limit law?

So I've learned that $\lim_{x \rightarrow 0}{\frac{\sin(x)}{x}} = 1$ is true and the following picture really helped me get an intuitive feel for why this is true I have been told that this limit ...
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1answer
46 views

How do I evaluate $\displaystyle\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.
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1answer
24 views

How to prove a complex limit with epsilon delta definition?

I have $$\lim_{z \to i} \frac {iz^3-1}{z+i}=0$$ To prove this I am trying to use the epsilon-delta definition. By saying that for any $\delta >0$ and any $\varepsilon >0$ then: ...
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1answer
12 views

Removing Discontinuity in 3-space without changing the partial derivative

Is it possible to find a version of the function $$f(x,y) = x\cdot \lfloor y \rfloor + \lfloor x\rfloor^2$$ That is continuous. ANY operation is allowed in changing the function as long as the ...
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0answers
44 views

Series $\sum \lambda^{n-k} c_k $ converges to zero

Let $(c_n)$ a sequence of real number, such that $\lim_{n \to \infty} c_n=0$, Let $0<\lambda<1 $ and $\lambda^nc_0+\lambda^{n-1}c_1+\cdots+\lambda c_{n-1}+c_n=y_n$ a sequence. I have to prove ...
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2answers
32 views

Limit of a sequence, possibly requires epsilon delta

Show that if $\{a_n\}_{n=1}^\infty$ and $\{b_n\}_{n=1}^\infty$ are sequences for which $\lim_{n\to\infty} a_n = 0$ and $\{b_n\}$ is bounded, then $\lim_{n\to\infty} a_nb_n=0.$ This is what I ...
4
votes
2answers
58 views

Check that $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$

Show that $$\lim_{n\to\infty}\sum_{i=1}^{n}\left(\frac{i+x}{n}\right)^n=\frac{e^{x+1}}{e-1}$$ Any hints how I can tackle this problem? Although I checked on a sum calculator that it converges ...
0
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1answer
33 views

How to evaluate the limit of this function?

If $f(x)=\bigg(1+\dfrac1x\bigg)^x,x>0$ then evaluate this limit:$$\lim_{n\to \infty}\bigg\{f(1/n)f(2/n)f(3/n)\dots f(n/n)\bigg\}^{1/n}$$ My attempt: i rewrote this as ...
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1answer
33 views

calculus question epislon delta [on hold]

let $f(x):R->R$ a function we know that the two following limits do exist : $lim_{x->+\infty}f(x)$ $lim_{x->-\infty}f(x)$ does $f(x)$ have max , min ? if so prove..
4
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1answer
62 views

Finding $\lim_{x \to 0}\frac{\sqrt[3]{1+2x}-1}{x}$

Good morning everyone, first time posting here. For my calculus class, we are asked to find $$\lim_{x \to 0}\frac{\sqrt[3]{1+2x}-1}{x}$$ We are given the hint that ...
2
votes
1answer
31 views

When are the limits of roots of a polynomial identical to the roots of the limit of the polynomial?

I have a univariate polynomial of degree $n$ (where $n$ is larger than $4$). The real-valued coefficients of the polynomial depend on a parameter $\psi$, i.e. $$p_\psi(x)=a_n(\psi) x^n+a_{n-1}(\psi) ...