Use this tag for questions on the concept of infinity. Don't use this tag merely because $\infty$ appears somewhere in the question.

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2answers
31 views

size of infinite strings and infinite alphabets

Please forgive the lack of formal vocabulary. Which set has a larger cardinality? A) a set of all possible countably infinite strings with a finite alphabet of symbols. B) a set of all possible ...
0
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2answers
26 views

The intersection of an infinite number of subspaces is a subspace

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite ...
0
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0answers
36 views

integral vs. residue at infinity

I have an issue with residues at infinity. I am computing the integral $\displaystyle{\int_{C_3^+(0)} \dfrac{e^{3z}}{z^2(z^2+2z+2)} dz} $ Since all three poles ($0$ of order 2, $1\pm i$ of order 1) ...
5
votes
4answers
121 views

Is $\aleph_1\cdot\aleph_1=\aleph_1$?

I'm currently trying to understand the basic notions concerning infinity. I think I understand that $\aleph_0\cdot\aleph_0=\aleph_0$ but how about $\aleph_1$? Is $\aleph_1\cdot\aleph_1=\aleph_1$ i.e. ...
4
votes
2answers
410 views

Compute a limit or prove that it does not exist

Do the following limits exist? Compute them or prove that they do not exist. (a) $\lim_{x\to 1}\frac{x^2-x}{2x^2-x-1}$ (b) $\lim_{x\to 1}\frac{|x-1|}{2x^2-x-1}$ For (a) it's pretty easy to see that ...
2
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2answers
38 views

Proving the following limit statements

I need to prove those If $f(x)\ge 0$ and $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}\sqrt{f(x)}=\sqrt{L}$. If $\lim_{x\to x_0}f(x)=L$, then $\lim_{x\to x_0}|f(x)|=|L|$. If $f(x)\ge g(x)$ for ...
1
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0answers
35 views

Friend B and C have eaten zero apples. How many more apples has C eaten?

Friend $A$ claims that he has eaten $1$ apple today. Friend $B$ responds. Congrats, I have eaten $0$ apples, so that is $\infty$ more apples than me. Friend $C$ says, but I have also eaten $0$ ...
1
vote
1answer
21 views

Probability of selecting a number in a repeating decimal series

For example in a infinitely repeating series such as $\frac{110}{111}=0.\overline{990}$, what would be the probability of selecting a 0 in the series generated by the infinitely repeating decimals? I ...
-2
votes
2answers
50 views

Limit of infinity times 0

I have a question regarding a specific step in the proof of the theorem that 'differentiability implies continuity'. The proof in my calculus book asserts that if $h\to0$ then: ...
1
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2answers
122 views

Is it true that the sum of all numbers equal 0? [closed]

I'm not a mathematician but I'm studying Nothing, so 0 is relevant, and I'm wondering about the fact that numbers seem to be mutually canceling polarities extending from 0, that is ...
0
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2answers
37 views

Determine limit of indeterminate form.

If the question is $$\lim_{x\to\infty}(e^x+1)^{\frac1x}$$ Do you just say that because $\lim_{x\to\infty}\frac1x$ is $0$, the original function has limit approaching 1, without caring the $e^x$?
2
votes
2answers
105 views

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional.

Suppose that $V$ is a vector space, and $W$ is a subspace of $V$. If $V$ is finite dimensional, then prove $W$ too must be finite dimensional. It seems intuitively obvious that the dimension of ...
0
votes
0answers
30 views

How to solve a particular indeterminate form

So the answer says $$\lim_{x\to \infty}x^2\sin\left(\frac1x\right)=\lim_{h\to 0^+}\frac1h\frac{\sin h}h$$ How does the transformation work?
3
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2answers
29 views

Limits and infinity in a succession

Apologies for this rather basic question. I am preparing the entry exam for university without the help of a teacher and occasionally get stuck on seemingly simple things. I have been all over the ...
1
vote
1answer
39 views

Proving using the squeezing theorem

The question is : I am not sure how i should make the function bigger and smaller in order to find the right limit for it. Thanks in advance !
1
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0answers
30 views

Complex residue at infinity of $f(z)=\frac{z^5}{\sin\left(\frac{1}{z^2}\right)}$

I'm having trouble finding residue of the function $$f(z)=\frac{z^5}{\sin\left(\frac{1}{\large{z^2}}\right)}$$ at infinity. Wolfram kindly informs that it is equal to $-\frac{7}{360}$ (and gives ...
1
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1answer
18 views

mass concentration inequality for polynomials

I am trying to prove the following: Let $p$ be a polynomial of degree n and let $I=[0,1]$ and $E\subset I$ a measurable set of non-zero measure, i.e., $\mu(E)\neq 0$. Then, $$\sup_{x\in I}|p(x)|\leq ...
0
votes
2answers
79 views

Number of elements is odd infinity [closed]

If we have one continuous function F(x), and if we define f(x)=F(x) on domain from open interval (a, b), and if F(a)=F(b) If function f(x) is monotonically increasing from point a to point M, and ...
-4
votes
1answer
38 views

Function approaches zero but derivative doesn't [duplicate]

If: $y=f(x)$ and $y=0$ when $x\rightarrow\infty$ Is it possible that: $\frac{d}{dx}(y)$ is not equal to zero when $x\rightarrow\infty$ And prove it!
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2answers
67 views

Showing that a set is countably infinite by defining a bijection between $\Bbb N$ and that set.

I'm a little confused on what is being asked here: Show that the following sets are countably infinite, by defining a bijection between $\Bbb N$ (or $\Bbb Z^+$) and that set. The set of positive ...
0
votes
3answers
33 views

proving that delta exists to a limit

The question is : Assume $\lim_{x\rightarrow1}f(x)=5$. Prove that there exists $\delta>0$ s.t. for every $x$ that sustains the condition $|x-1|<\delta$, we know that $f(x)>-1$. I know ...
1
vote
2answers
65 views

Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? [duplicate]

From what I understood, cardinal numbers are defined as: $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$ The continuum hypothesis ...
2
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1answer
26 views

Linear independence and ∞-dimensioned vector spaces

I have a question regarding this problem: Let ℝℕ be the vector space of all infinite real sequences. Show that even though its infinite subset X≔{(1,0,0,…), (0,1,0,0,…), (0,0,1,0,…)} is linearly ...
1
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1answer
37 views

Once we've constructed N, do we need again the axiom of infinity to conclude the set of the primes is infinite?

When completing the proof (unfortunately, at the same time presenting it as Euclid's and performing it ad absurdum) of the infinitude of prime numbers, my algebra professor stated "...and thus the set ...
2
votes
3answers
73 views

How do I compute $\sum_{k=1}^{\infty} k \cdot p^k$ [duplicate]

I have no idea how to compute this infinite sum. It seems to pass the convergence test. It even seems to be equal to $\frac{p}{(1-p)^2}$, but I cannot prove it. Any insightful piece of advice will be ...
1
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1answer
58 views

Infinite lines and points

Ok, lately I have been thinking a lot about one idea that has been bothering me since first I learned about lines and points. I understood that: A line has no thickness, is straight and it is ...
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votes
1answer
145 views

Why does this mathematics Professor believe -1/12 is a mysterious value related to 'infinity'? [duplicate]

In the following YouTube video, a maths Professor talks about -1/12 and its mysterious relationship with infinity: "Why -1/12 is a gold nugget" He says, about the sum of the powers of natural ...
0
votes
1answer
31 views

Can infinity be represented as a recursive function?

Could infinty be represented as a recursive function like: The function f takes any number x as parameter and returns f(x+1), resulting in an endless recursive call, each call incrementing x by 1. ...
2
votes
1answer
76 views

What does this product converges to?

Let $p\in[0,1]$. I'm interested in computing $$\lim_{n\to\infty}\prod_{i=1}^n(1-p^i)$$ Any thoughts? EDIT: As Kibble mentioned, this is the Euler function. Also from Kibble: a simple upper ...
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6answers
52 views

how to find an infinity limit in a fraction

I don't understand how to find the limits of this expression when $x\to\infty$ and $x\to-\infty$: $$\left(\frac{3e^{2x}+8e^x-3}{1+e^x}\right)$$ I've searched for hours. How to compute these limits?
3
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3answers
92 views

$f(x)$ has a limit, prove that $\sqrt{f(x)}$ has a limit

The question is : Let $f$ be a positive function defined on an interval $[a,\infty)$, such that $\lim\limits_{x\to\infty} f(x)=0$. Prove that $\lim\limits_{x\to\infty} \sqrt{f(x)}=0$. If $f(x)$ ...
2
votes
1answer
30 views

Proving a limit on a general function

The question is : Not sure how to even start, i know that |F(x) - L| < Epsilon so if F(x) = L then the function is liner on L, Don't know how to prove its an integer. Thanks in advance !
1
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2answers
38 views

Solve using the limit's definition

The question is : I am stuck here : $$| \sqrt{x+1} - \sqrt{x} -1 | / \sqrt{x} + 1$$ i know that the numerator is negetive so i must change it in order to delete the absolute value yet i still ...
0
votes
2answers
31 views

Finding a very close epsilon

This is the Question : Given the following limits, find $M>0$ such that for every $x>M$ the expression are $\frac13$-close to their limit (in other words, find $M>0$ s.t. for every ...
0
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1answer
87 views

Proof that √x does not tend to a limit as x approaches infinity

I am wanting to prove from the definition of a limit ( $∀ε>0 ∃K>0:∀x>K, |f(x)−l|$<ε) that $√x $ does not tend to a limit as $x$ approaches infinity. So far I have tried to find a value of ...
1
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1answer
57 views

limit as x approaches zero from left and right equals positive and negative infinity

How can we prove \begin{align} \lim_{x\to 0^+}\frac1x &= +\infty, \\ \lim_{x\to 0^-} \frac1x &= -\infty \end{align} This seems really simple but I'm having trouble starting it
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2answers
70 views

Infinity In Math - The Nick Lim Proposal [closed]

So infinity is clearly a very strange, concept. So I have the following proposal (the nick lim proposal) which can not be solved ( at least to my current knowledge, hopefully you can shed some light ...
3
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0answers
36 views

The behavior of the 3D wave equation close to the origin

The general solution to the three dimensional wave equation is \begin{equation}u(r,t) = \frac{F(x+ct)}{r} + \frac{G(x-ct)}{r} \end{equation} where $F$ and $G$ are arbitrary functions. I want to ...
0
votes
3answers
47 views

What is the limit of the following sequence? $\lim_{n\to\infty} 8^\frac{n+1}{3n+2}$

What is the limit of the following sequence? $$\lim_{n\to\infty} 8^\frac{n+1}{3n+2}$$ I substitute infinity in $n$ and I get infinity + 1 = infinity, 3*infinity+2 = infinity. Infinity over infinity = ...
0
votes
2answers
50 views

Why is the limit of this function tending to 1?

$$ \lim_{x\to \infty} \left(\frac{1}{(x^2+x)\left(\ln\frac{x+1}{x}\right)^2}\right) $$ I know the answer is 1, but why does it tend to 1? Can you manipulate the function and the "$\ln$" to make it ...
1
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1answer
36 views

infinity over infinity and zero multiplied infinity in a calculation which gives (correctly) 1

I hope you don't put as a duplicate my question as it is thought for the specific case I am going to show you now, which is about a calculation. I read about the topic of infinity over infinity, ...
4
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2answers
438 views

Uncountable vs Countable Infinity

My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity. As far as I understand, the list of all natural numbers is ...
0
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0answers
35 views

Definition of the rank of infinite matrix

How is the rank of an infinite matrix defined? Is it the same as in the finite case, i.e. the number of elements in a basis for some matrix? How are the dimensionalities of the column and null ...
1
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0answers
22 views

Is it formally correct to take the limit of infinitely many terms?

Let us take the most basic example: $\lim_{n\to\infty} \frac{n!}{n^n}$ $\frac{n!}{n^n} = \frac{n(n-1)...1}{n \cdot n...n} = \frac{n}{n} \cdot \frac{n-1}{n} \cdot [...] \cdot \frac{1}{n} \leq ...
4
votes
5answers
87 views

Why can't $\int_{-1}^1{\frac{dx}{x}}$ be evaluated?

So I just saw this question on brilliant.org and found that many people argued over whether this could be evaluated. My answer was 0 because it is an odd function, but others argued that because of ...
2
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2answers
65 views

How to write down proof that if $\lim_{x\to \infty}f(x)=\alpha$ then $\lim_{x\to \infty}f'(x)=0$?

Let $a, \alpha \in \Bbb{R}$; let $f: (a,+\infty)\to \mathbb{R}$ be differentiable; let $\lim_{x\to \infty}f(x)=\alpha$; let $\beta := \lim_{x\to \infty}f'(x)$. I want to show that $\beta = 0$. Now, ...
0
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2answers
22 views

Evaluation of this limit

I am looking in a calculator how to evaluate the limit as x approaches infinity for the following function: $\sqrt{x} -\sqrt{x-1}$. In the evaluation the calculator applies the following algebraic ...
0
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2answers
50 views

Sizes of infinities

How is it that some infinities are bigger than other infinities and also If I have an infinite amount of apples and an infinite amount of planet earths then which will have the greater mass? (My ...
0
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4answers
55 views

Special limit: $1^\infty$

This is my limit, $\lim\limits_{n \to \infty } (1+\frac{1}{n^2})^{3n^2+4}$. If I place the infinity instead of $n$, I get $1^\infty$. I know that this is a special limit, but how I need to ...
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1answer
42 views

Proving Limit is a sequence of positive numbers

$$\lim_{n \rightarrow \infty}{ a_n=0}{\implies \lim_{n \rightarrow \infty}{\frac{1}{a_n}=\infty}}$$} Assuming this sequence is of positive numbers, can anyone give any tips how to start proving this? ...