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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2
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2answers
77 views

Can I assume the continum hypothesis in a proof

I am showing that the cantor ternary set has the same cardinality as $\mathbb{R}$ I want to use the fact that it is uncountably infinite and a subset of $\mathbb{R}$. ($|N| < |C| \leq \mathbb{R}$) ...
-1
votes
2answers
78 views

Limit laws when not both limits exist

In the calculus textbooks I've come across, the limit laws are given on the condition that both individual limits exist. Is it safe to weaken that condition by saying that they are valid as long as ...
2
votes
1answer
74 views

There are more languages than programs?

I am reviewing some Turing machine material...and I come across this the set of all programs are countable (convert them into binary string, each of which represent an integer) whereas the set of ...
3
votes
1answer
43 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
1
vote
3answers
73 views

Proof using formal definition: Infinite limit

I was wondering how get the proof of this limit: $$\lim\limits_{x\to -\infty}\dfrac{{x^2} - x + 1}{x + 4} = -\infty$$ The problem is that I don't know what to do for find the appropriated values to ...
0
votes
3answers
73 views

probability on countable infinite sets

My question relates to probabilities on countable infinite sets. For example, what is the probability of choosing an even number from the positive integers. Believe it or not I am interested in this ...
0
votes
1answer
55 views

Thomson's Lamp Question

The Thomson's Lamp paradox: A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a ...
0
votes
3answers
63 views

calculate the limit of the following function.

$$\lim_{x \to 0} \frac{5x - e^{2x} + 1}{3x + 3e^{4x} - 3}$$ For some reason I can't find the trick to solve this. Tried a lot but it always come to a place where its $\frac{0}{0}$ or ...
0
votes
0answers
49 views

evaluate $\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$ [duplicate]

I am attempting to evaluate the following integral: $$\int_0^\infty \frac{(ln(x))^2 }{1+x^2}dx$$ Using the substitution $x=e^u$ and $dx=e^u du$, I get: $$\int_{-\infty}^\infty \frac{u^2}{e^{-u} + ...
-2
votes
1answer
33 views

Zero-infinity hypothesis [duplicate]

math.stackexchange community. I have joined to inquire on a hypothesis a friend of mine has recently proposed. Please note: before posting this, I have repetitively told him that his logic is flawed ...
0
votes
2answers
33 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
votes
2answers
46 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
votes
0answers
46 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
124 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
437 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
votes
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
vote
0answers
36 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
24 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
vote
2answers
142 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
votes
2answers
38 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
168 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
32 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
votes
2answers
37 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
41 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
vote
0answers
37 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
vote
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
82 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
45 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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votes
2answers
84 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
70 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
votes
1answer
28 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
vote
1answer
38 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
77 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
vote
1answer
62 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 ...
-8
votes
1answer
318 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
34 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
86 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 ...
1
vote
6answers
53 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
votes
3answers
97 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
vote
2answers
39 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
votes
1answer
108 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
vote
1answer
79 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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votes
2answers
77 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
37 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
49 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
51 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 ...