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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.

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15 views

Prove this formula follows from a function being continuously differentiable?

I'm studying for an exam in an electrical engineering course (stochastic process in dynamic systems), though this section is strictly on the math. A given practice problem (with no solution given, of ...
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1answer
52 views

Find the limit of $\lim_{x\to\infty}\left(\frac{{(\sin(x))^{2}}{}}{{3}^{x}}\right)$

I am having trouble trying to solving this limit. I can clearly see that answer is $0$, but I do not know how to deal with $sin(x)$ because it has no definitive limit. $$\lim_{x\to\infty}\left(\frac{...
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0answers
12 views

The monotonicity of a weighted entropy

Now we have $$p(x) = \frac{d!}{x!\,(d-x)!\,2^d}$$ $$e(x)=\frac{x}{d}\log\Bigg(\frac{d}{x}\Bigg) + \frac{d-x}{d}\log{\frac{d}{d-x}}$$ and $$E(d) = \sum_{0 \leq x \leq d}{p(x)e(x)}$$ How to prove the ...
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2answers
30 views

determine if the statement with limit is true

I want to determine if the statement $$\lim_{x\to\ a} f(x) = \infty \Rightarrow \lim_{x\to\ a}\frac{1}{f(x)} = 0$$ is true or not (by proving it or proving a contradiction). I know that I have a ...
1
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2answers
48 views

Prove that a point $x$ is an accumulation point of $A$ iff $x$ is in $\overline{A\setminus{\{x}\}}$ in a Hausdorff topological space. [on hold]

As the title says, I'm trying to answer a question which asks me to show that, for a non-empty subset $A$ of a Hausdorff topological space, a point $x$ is an accumulation point of $A$ (i.e. $x \in A'$)...
2
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1answer
43 views

Proof validation $\lim_{n\to\infty} \left(1+\sum_{k=1}^n \frac{(-1)^k}{k!}\right) = {1\over e}$

Prove the following limit: $$ \lim_{n\to\infty} \left(1+\sum_{k=1}^n \frac{(-1)^k}{k!}\right) = {1\over e} $$ The instrumentation I have so far is: $$ \lim_{n\to\infty}\left(1+{1\over n}\right)^n =...
3
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2answers
30 views

convergence of a recursive sequence with parameter a

How can you determine if the following recursive sequence converges: $$x_{n+1}=\frac{1}{2}(a+x_n^2)$$ where $0\le a \le 1$ and $x_1=0$ I know that the limit x (if it exists) satisfies the following ...
2
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3answers
60 views

Show that $\lim_{n\rightarrow\infty}e^{-n}=0$

I wrote out the term from above and get $\text{Show that: }\lim_{n\rightarrow\infty}\sum_{k=0}^{\infty}(-1)^k\frac{n^k}{k!}=0$ I can use Leibniz' Criterion to find out that it converges but I don't ...
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3answers
68 views

Show that $\lim_{h\rightarrow 0}\frac{e^{ih}-1}{h}=i$

$h\in \mathbb{R}$, because we have defined the Trigonometric Functions only on $\mathbb{R}$ so far. I have a look at $e^{ih}=\sum_{k=0}^{\infty}\frac{(ih)^k}{k!}=1+ih-\frac{h^2}{2}+....$ How can ...
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0answers
23 views

Why do the properties of $\lim_{n\rightarrow\infty}(x_n)$ still hold for $\lim_{x\rightarrow x_0}f(x)$

Where is the Transition of sequencences and functions? With properties I mean for example rules like the sandwichtheorem $$\lim_{n\rightarrow\infty}a_n=K, \lim_{n\rightarrow\infty}c_n=K\text{ and }...
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2answers
44 views

Compute $\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$

Compute $\lim_{n \to \infty}(1+\frac{n-1}{n+1})^{\frac{n+1}{n-1}}$ I did: $\lim_{n\to \infty}(1+\frac{1}{\frac{n+1}{n-1}})^{\frac{n+1}{n-1}}=e$. Why is this incorrect? Thank you for your help.
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3answers
45 views

Can any unbounded sequence $ (a_n) $ has a representation as $ a_n=c_n b_n $ where $ (c_n) $ is a bounded sequence, $ b_n $ tends to infinity?

Can any unbounded sequence $ (a_n) $ has a representation as $a_n=c_nb_n$ where $ (c_n) $ is a bounded sequence, $ b_n$ tends to infinity ?
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1answer
29 views

Problem regarding Epsilon-Delta definition of limits

I came across this problem in "Introduction to Real Analysis" by Bartle and had a doubt regarding the solution to a given problem. I needed to determine a condition on $|x-1|$ that will ensure that $$...
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1answer
30 views

Proving a limit exists using delta and epsilon?

First time posting, I did calculations, would someone be so kind to see if i'm right or wrong, i'm a part-time student so i do not have a lecturer etc. f (x,y) = x2/ √(x2+ y2) Prove from First ...
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3answers
56 views

limit as $x$ approaches infinity of $(\ln(x+1)/\ln(x))^x$ [on hold]

I've been told splitting the limit into the product of $x$ limits would make it an indeterminate form, and applying L Hopitals rule seems tricky too. Thanks in advance.
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0answers
31 views

Any Contradictions to My Axiom About Infinity Arithmetic? [on hold]

I have been reading up several different questions and responses to questions about infinity, and most of the responses are very valid with the standard assumptions of mathematics, but others (though ...
4
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2answers
91 views

How to prove that $\lim_{n\to\infty} \sqrt[n]{a^n+b^n}=\operatorname{max}(a,b)$? [duplicate]

How do I show that $$\lim_{n\to\infty} \sqrt[n]{a^n+b^n}=\operatorname{max}(a,b)$$ with $a,b\ge0$. I tried to do this by dividing it in two cases, when $a=b$ and $a\gt b$. In the case $a\gt b$ I ...
1
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2answers
87 views

Assuming that $f''(a)$ exists, show that: $f''(a)=\lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$ [duplicate]

Assuming that $f''(a)$ exists, show that: $$f''(a)=\lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$$ I got the first derivative by using the limit rule $(f(a+h)-f(a))/h$. But when taking the second ...
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1answer
40 views

Find $\lambda$ such that $f$ it is differentiable in zero and has a continuous derivative in zero

I am trying to solve this task Find $\lambda>0$ such that $f=\begin{cases}0& x=0\\ |x|^{\lambda}\cdot \sin\frac{1}{x} & x\neq 0 \end{cases}$ a) is differentiable in zero b) ...
0
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1answer
48 views

Does Cauchy second limit theorem work both ways? [duplicate]

I have been told, that Cauchy second limit theorem doesn't always work both ways - if the limit of $n$th root of sequence exist it doesn't immediately mean the limit of $a_{n+1}/a_n$ is equal to nth ...
1
vote
1answer
31 views

Uniform convergence of $f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$

I need to study the uniform convergence of $$f_n(x) = \frac{nx}{(2+nx)(4+x^2)}$$ on the interval $[2,+\infty)$ I've shown that on $[0,+\infty)$: at $x =0$ $f_n(0)=0 \xrightarrow{} 0$ at $x \neq 0$...
0
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1answer
21 views

Uniform convergence of $f_n(x) = n(x-1)e^{-nx}$

I need to study the uniform convergence of $f_n(x) = n(x-1)e^{-nx}$ on the interval $[0,+\infty)$ I've shown that : at $x =0$ $f_n(0)=-n \xrightarrow{} -\infty$ at $x =1$ $f_n(1)=0 \xrightarrow{} ...
1
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2answers
39 views

Write a Limit to calculate $f'(0)$

Let $f(x) = \frac {2}{1+x^2} $ I need to write a limit to calculate $f'(0)$. I think I have the basic understanding. Any help would be greatly appreciated. d=delta and so far what I have is $f'(...
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2answers
22 views

Use Limits to calculate slope of the tangent

Use limits to calculate the slope of the tangent to the curve $y=\frac1x$ at $x=a$. I need to write an equation for the tangent to $y=\frac1x$ at $x=4$. I think I understand the basics of the ...
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2answers
36 views

$ \lim_{x\to 0} \left(\frac{5}{2+\sqrt{9+x}}\right)^{\operatorname{cosec} x}$

I solved this question in two different ways and got two different answers to it. Method #1: The question is of the form $1^\infty$. So, to solve it, I took it in the power of e and wrote the ...
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2answers
62 views

Finding limit of $\sin(x^2/2)/\sqrt{2}\sin^2(x/2)$ as $x\rightarrow0$

Can anybody help me find the limit as $x$ tends to $0$, for $$\frac{\sin(x^2/2)}{\sqrt{2}\sin^2(x/2)}.$$ How can I simplify the expression or use equivalent transformations to find a limit (without ...
3
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2answers
66 views

Evaluating $\lim_{x\to 1}\frac{^5x-^4x}{(1-x)^5}$, where $^nx$ is the repeated exponent (“tetration”) operation

$$\lim_{x\rightarrow1}\frac{x^{x^{x^{x^x}}}-{x^{x^{x^x}}}}{(1-x)^5}$$ My friends told that it appeared on Instagram (some social media network). I tried various methods but failed. (I tried using ...
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1answer
33 views

Finite limit for a function that becomes infinite.

I have a function like this: $$f(x) = \Big(1-\frac{1}{\cos x}\Big)\frac{1}{\sin x}$$ I need to evaluate the limit of this function when $x\rightarrow \pi /2$. A simple calculation shows that the ...
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0answers
23 views

Equivalent condition to the existence of a limit, help understanding proof

This is part of problem 5-23 in Michael Spivak's Calculus; I changed the proposition to be proven in order to omit what is already clear to me (the proof of the implication in the other direction). I ...
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2answers
50 views

How can I calculate this limit? (With Riemann sum ?) [on hold]

How can I calculate this limit? $$\lim_{n\rightarrow\infty} \frac{ \sum^{n}_{k=1} \cos(k\cdot \frac{\pi}{n})}{n}$$ I thought about Riemann sum , but, I don't know how to do it.
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3answers
51 views

Find the convergence radius of \sum_{n=1}^\infty {\frac{(2n)!}{(n!)^2}}z^n

I need to find the convergence radius of $$ \sum_{n=1}^\infty {\frac{(2n)!}{(n!)^2}}z^n $$ I proceeded like this: $$ \lim \sum_{n=1}^\infty {\frac{(2n+2)!}{((n+1)!)^2}}z^{n+1} \times {\frac{(n!)^2}{(...
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2answers
79 views

How to find limit $\lim_{n\to \infty}\left[(n+1)\int_{0}^{1}x^n\ln(x+1)dx\right]$ [duplicate]

How to find this limit; $$\lim_{n\to \infty}\left[(n+1)\int_{0}^{1}x^n\ln(x+1)dx\right]$$ What techniques can I use here? Thanks for reading and help. This question was asked in GATE 2008.
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2answers
90 views

How to find the limit $\lim_{n\to\infty} {2^n+(-1)^n \over 2^{n+1}+(-1)^{n+1}}$?

How do I find this limit? $$\lim_{n \to \infty} {2^n+(-1)^n \over 2^{n+1}+(-1)^{n+1}}$$ I don't know how to take the limit of $\lim_{n \to \infty}{(-1)^n}$, I know that the limit $\lim_{n \to \infty} {...
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3answers
36 views

Evaluating big-O vs big-Omega for two functions

We were tasked with comparing the complexities of two functions: $n$ and $n^{0.99} (\log(n))^2$. As I understand it, the general construct of these equations is $\lim_{x \rightarrow \infty} \frac{f(...
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1answer
33 views

Show that a function is monotone and its set of discontinuities is equal to a sequence.

Let $(a_n)_{n\geq 1}$ be a sequence with $a_n \in [0,1]$ for all $n \geq 1$. Show that $$f\colon[0,1]\to [0,1], \quad f(x) = \sum_{\{k : a_k\leq x \}}2^{-k-1}$$ is a monotone function and that the ...
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0answers
38 views

Integral from -infinity to infinity as a pair of distinct integrals

In my class we have said that an integral from -infinity to infinity of a function f exists, when a point c exists, such that both improper integrals (Integral from -infinity to c of f) and (Integral ...
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1answer
47 views

limit of a sequence with parameter [on hold]

Can somebody help me to find the limit of the next sequence, please? $$x_0=a>0, x_{n+1}=a+\frac{1}{x_n};n>0 $$
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2answers
32 views

How to solve the limit $\ \lim_{\substack{x\to 0 \\ y\to 0}}\sin(xy)/y\ $? [on hold]

Good day. I have tried solving this limit by using some methods, such as L'Hopital Rule, notwithstanding I did not get nothing substantial.
3
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3answers
32 views

Limits with Taylor series around zero

I had some problems with the following two limits, which are supposed to be calculated with Taylor series: $$ \lim_{x\to 0^+}\frac{e^\sqrt{x}-e^{-\sqrt{x}}}{\sqrt{\sin{2x}}}\quad\mbox{and}\quad \lim_{...
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2answers
43 views

Show by means of an example that $\lim_{x \to a} [f(x)g(x)]$ may exist even though neither $\lim_{x \to a} f(x)$ nor $\lim_{x \to a} g(x)$ exists. [on hold]

This is particularly a hard problem to solve, "a" has to be a defined number. We can't really pick any two functions simply like $\frac{1}{x}$ and any other to exemplify. Please help me with it
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3answers
52 views

prove that $\lim_{n\to \infty} \sqrt[n]{\frac{n^{(n^2+1)}}{(n+1)^{n^2}}}=\frac 1e$ [on hold]

I tried to solve that limit, but I wasn't able to see any solution: $$\lim_{n\to\infty} \sqrt[\large n]{\frac{n^{(n^2+1)}}{(n+1)^{n^2}}}=\frac 1e.$$ Does anyone have an idea?
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2answers
47 views

Given $S_n = 1 + \sum_{k=1}^n {1\over k!}$ prove that $e-S_n \le \frac{n+2}{n!(n+1)^2}$

Let $S_n$ be a sequence defined by: $$ S_n = 1 + \sum_{k=1}^n {1\over k!} $$ Prove that: $$ e - S_n \le \frac{n+2}{n!(n+1)^2} $$ This problem comes in the limits section so i may use anything ...
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votes
1answer
17 views

How to prove that this infinite sum converges?

$a_1,...,a_k$ are real numbers all bigger then $1$. Define the following sequence of numbers: $$b_n=\frac{1}{\sqrt[n]{a_1^{n^2}+a_2^{n^2}+...+a_k^{n^2}}}$$ Show that $\sum_{n=1}^{\infty} b_n$ ...
0
votes
1answer
31 views

Evaluating $\lim_{x \to 0} \frac{a_1\exp(-b_1x^2)}{\sum_i a_i\exp(-b_ix^2)}$

I am looking for the solution of the following limit: $$\lim_{x \to 0} \frac{a_1\exp(-b_1x^2)}{\sum_i a_i\exp(-b_ix^2)}$$ Since $\lim_{y\to0}\exp(y) = 1$, is $\frac{a_1}{\sum_i a_i}$ really the ...
0
votes
0answers
19 views

Calculate the limit of special sequence.

There is given a sequence $A_{0},A_{1},...$ wich are all ordered zeros of Mertens fuction $M(n)$. How to find the following limit ?: $$\lim_{n\to\infty}\frac{A_{n+1}-A_{n}}{A_{n}^{0.75}}$$ Does ...
1
vote
1answer
27 views

For polynomial $g(x)$ satisfying $(g(a))^2+(g'(a))^2=0$, evaluate $\lim_{x\to a}\frac{g(x)}{g'(x)}\left\lfloor\frac{g'(x)}{g(x)}\right\rfloor$

If $g(x)$ is a polynomial function and $$(g(\alpha))^2+(g'(\alpha))^2=0$$ then evaluate $$\displaystyle \lim_{x\rightarrow \alpha}\frac{g(x)}{g'(x)}\bigg\lfloor \frac{g'(x)}{g(x)}\bigg\...
4
votes
4answers
56 views

Compute $ \lim\limits_{n \to \infty}\frac{\sqrt{3n^2+n-1}}{n+\sqrt{n^2-1}}$

Compute $$ \lim\limits_{n \to \infty}\frac{\sqrt{3n^2+n-1}}{n+\sqrt{n^2-1}}$$ I did the following: $$ \lim\limits_{n \to \infty}\frac{\sqrt{\frac{3n^2}{n^2}+\frac{n}{n^2}-\frac{1}{n^2}}}{\frac{n}{n^...
0
votes
2answers
43 views

Relationship with derivative

Let $F :[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that it’s derivative $F’(x)$ is increasing in $x$. Then which of the following is true for every $x$,$y$ in $[0,1]$ with $x>y$....
0
votes
0answers
30 views

Limit of Matrix power equals $\pi$ [on hold]

Denote matrix \begin{equation} A= \begin{pmatrix} 1-t&2-t\\-t&1-t \end{pmatrix} \qquad (0<t<1) \end{equation} and vector $u_n\equiv(u_{1n},u_{2n})^\top=A^n v$, where $v=(1,1)^\top$. $N(t)...
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votes
2answers
39 views

Showing that a sequence is convergent, and finding its limit [on hold]

Okay, so I am very confused about these particular types of problems. Now, I have seen analysis before, so I think that I am just making this way more difficult for myself. I need to show that a ...