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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43
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11answers
3k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
5
votes
4answers
204 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
1
vote
4answers
217 views

why $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)=\infty$?

Why does $\lim\limits_{x\to-\infty}(\sin x+2)\ln(-x)$ equal $\infty$? Breaking up the limit: $\lim\limits_{x\to-\infty}(\sin x+2)$ DNE because it oscillates between 1 and 3 $\lim\limits_{x\to-\...
0
votes
0answers
20 views

Random walk on a segment with infinite time

Given a point particle on a segment $L$ of length $1$, $(L=[0,1])$, assume the particle moving randomly in such a way: $p_{(k+1)}=p_k+\delta_k$ where $p_{k+1}$ is the position on the segment at time $...
2
votes
1answer
58 views

A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
0
votes
1answer
35 views

Compute H-infinity norm in Matlab

Please can someone write a command in Matlab for calculating $H_{\infty}$ norm for the following system: $$\frac{d}{dt}z(t)=Az(t)+Bu(t)+Fw(t)$$ $$y(t)=Cz(t)+Du(t)$$ where $A$, $B$, $C$, $D$, and $F$ ...
1
vote
1answer
36 views

Inverse of a matrix with main diagonal elements approaching infinity

Let $A$ be a invertible, symmetric, positive definite $p \times p$ covariance matrix with main diagonal elements $a_{ii},~i = 1,~\ldots,~p$. If all main diagonal elements would approach $\infty$, ...
6
votes
1answer
837 views

Can a curve be an asymptote?

$f(x)=x^3+\frac{3}{x-1}$ This was the question given to me. I replied that $f(x)$ will have only a single vertical asymptote of $x=1$. My teacher told that there'll be be two asymptotes. One is the ...
43
votes
16answers
3k views

Different sizes of infinity

Correct me if I'm wrong, but this is what they taught us in precal: $$\lim_{x\rightarrow\infty}x=\infty$$ $$\lim_{x\rightarrow\infty}x^{2}=\infty$$ But, we also know that $n^{2}>n$ if $n\notin [0,1]...
2
votes
1answer
96 views

What's so different about limits compared to infinitesimals?

If you find the limit is 2 for a given function, wouldn't this be the same as $2 + \epsilon$ with $\epsilon$ being a negligible value? This different way of defining limit-like behavior seems rigorous ...
0
votes
3answers
106 views

How $\infty=\infty$.

If we contruct two strainght lines as shown: Then join them such that to complete a triangle. It is taught that we can find infinity points on straight line. So there are infinity points on $DE$ ...
3
votes
5answers
1k views

How to prove the infinite number of sides in a circle?

I was in geometry class today when I came across the following formula for the external angle of a regular polygon with n sides: $$Ea = \frac{360º}{n}$$ So I thought if $$ n\rightarrow\infty $$ then $...
4
votes
1answer
424 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
10
votes
6answers
11k views

Prove that the distance between a black and a white dot is one

I just read this article about some tough interview questions. One of the questions (allegedly given in an interview for a Technology Analyst position in Goldman Sachs) was: There are infinite ...
2
votes
2answers
58k views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
2
votes
1answer
67 views

Topology and Borel sets of extended real line

Let $\mathcal{B}_{X}$ denote the Borel $\sigma$-algebra on $X$. I'm reading a book on real analysis by Folland and he defines $$\mathcal{B}_{\overline{\mathbb{R}}} = \{ E \mid E \cap \mathbb{R} \in \...
2
votes
1answer
87 views

Proof of Lemma 8.2.3 in Terence Tao Analysis 1 book

$\textbf{Lemma 8.2.3 }$ Let X be a countable set, and let $f:X \rightarrow R$ be a function. Then the series $ \sum_{x \in X} f(x)$ is absolutely convergent if and only if $$ sup \left\{ \sum_{x \...
0
votes
1answer
51 views

What are the limit points of $A_n=[n,\infty)$ in a metric space? Is $A_n$ closed?

$A_n=[n,\infty)$ in $\mathbb{R}$ with a Euclidean metric. A set is closed if it contains all its limit points. A limit point is a point whose neighborhood contains a point in the set. I'm not sure ...
0
votes
2answers
32 views

Infinite differentiability with a removable discontinuity?

I'm still a beginner with calculus. But this puzzled me. Let's say you had $f(x) = \frac{x^2-1}{x+1}$. It's discontinuous at one point. If you took the derivative infinitely many times, would the ...
2
votes
1answer
53 views

Is this sequence going to infinity, and how do we know that?

$a+\dfrac {a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}} {b}=?$ I've tried letting $\quad a+\dfrac {a+\dfrac {a+\dfrac {:} {b}} {b}} {b}=K$ Which makes the equation: $a+\dfrac {K} {b}=K$ $\quad$ ...
3
votes
2answers
410 views

Why does Michio Kaku say that $\frac{1}{0} = \infty$?

Why does Michio Kaku say that $\frac{1}{0} = \infty$? http://youtu.be/AJ4zlvqOtE8?t=4m43s Instead of $\frac{1}{0}$ that's not defined, so we don't know.
4
votes
5answers
151 views

Why can't we just say that $\infty-\infty$ equals zero?

Let be $\lim\limits_{x\to \infty}x=A$ and $\lim\limits_{y\to \infty}y=B$. Can be $A-B=0$? If the answer is "no" , why? And my other example: $\displaystyle\int_{-\infty}^{\infty} \dfrac{x}{...
-10
votes
2answers
360 views

How is the set of natural numbers countably infinite. [closed]

Here is my question TLDR. How can the natural number system be considered "countably infinite", when it has subsets that are infinite within itself. Isn't a countably infinite set one that contains ...
0
votes
3answers
62 views

Where did $\sqrt{x^2/x^2}$ come from in $\lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty}\frac{-1-1/x}{\sqrt{x^2/x^2}} = -1$?

I'm reading a calculus book and I saw the following limit solution. $$ \lim_{x \to -\infty}\frac{x+1}{\sqrt{x^2}} = \lim_{x \to -\infty} \left(\frac{x+1}{\sqrt{x^2}} \cdot \frac{-1/x}{-1/x}\right) = ...
2
votes
1answer
53 views

What is meant by $\lim_{x\to \infty^+}$

I am familiar on how limits work and such. For example, look at the following limit: $$\lim_{x\to 5^+} \frac{-x^2+5x}{5-x} = 5$$ It is saying that, as $x$ approaches $5$ from the right, the equation ...
33
votes
10answers
4k views
2
votes
2answers
76 views

Can the extended real number $+\infty$ be compared to transfinite numbers such as $\aleph_0$?

If not, why not? If so, is ∞ greater than or less than $\aleph_0$? Edit: the discussion in comments (including comments on a deleted answer) have made me think that the best way to put the issue is ...
0
votes
2answers
41 views

Can one non-cardinal infinity be greater than other non-cardinal infinity?

As far as I know, there are two different notions to the word "infinity" in Mathematics. First notion of infinity has to do with the cardinality of a set: if a set contains infinite number of ...
1
vote
1answer
82 views

Set theory with multiple countable infinities [closed]

In set theory, all sets that are countably infinite are generally considered to have the same size since there is a bijection between them. Has anyone tried formalising set theory in a way which ...
0
votes
1answer
32 views

Voronoi edges example

I have 4 line segments: 0 0 2 0 // 1st line segment 2 0 2 1 // 2nd line segment 2 1 0 1 0 1 0 0 and I wrote some CGAL code to print the Voronoi edges. However, <...
47
votes
13answers
31k views

I have learned that 1/0 is infinity, why isn't it minus infinity?

My brother was teaching me the basics of mathematics and we had some confusion about the positive and negative behavior of Zero. After reading a few post on this we came to know that it depends on the ...
-3
votes
2answers
63 views

If $\dfrac{1}{\infty}=0$ then I can prove that $0 = 1$ [closed]

Given, $\dfrac{1}{\infty}=0$, then $1=0 \cdot \infty = 0$ (because $0$ times any number or values is $0$ and here that number is infinity). Which gives us $1=0$ i.e, $0=1$. Hence proved....
3
votes
5answers
874 views

How many points in a line segment?

My teacher said that in the circumference of circle there are infinite points. When I was learning more about circle, I came to this picture: My question is: When we unroll the circle, then the ...
0
votes
0answers
37 views

Is there an infinity smaller than countable? [duplicate]

In other words: is $\aleph_0$ the smallest infinity? Is it easy to prove?
-1
votes
2answers
75 views

Random Room changing in the Hilbert hotel. [closed]

Let's say you have a Hilbert's grand hotel full occupancy. Assign each occupant a new room select randomly without regard to whether the room is assigned to someone. i.e. empty rooms, multiple ...
4
votes
0answers
47 views

Flea on the coordinate system

We drop a flea on a point of the coordinate system(with integer coordinates). Due to the dimensions of the flea we can not see it. The flea jumps away every second by one unit (always in the same ...
0
votes
0answers
29 views

Complex variable limit at infinity

Is $\lim\limits_{z\to\infty} \frac{4z^2}{(z-1)^2}$, $z\in\mathbb{C}$, evaluated the same way as a real variable function limit? Or does one need to show separate cases for $x\to\infty$ and $y\to\infty$...
1
vote
1answer
46 views

Limit of $(-1/2)^n$ as $n$ approaches infinty

I tried plugging bigger and bigger $n$'s into my calculator and the result obviously approaches $0$ (albeit oscillating between positive and negative). So how do you prove that: $$\lim_{n \to \infty}...
1
vote
3answers
50 views

Arithmetic Operations with Infinities in Real Analysis

Infinity is not a number , thus we cannot perform the usual arithmetic operations that we do with real numbers This is the usual reason given when asked why we can't perform the usual arithmetic ...
0
votes
0answers
41 views

Difficulty in understanding Cantor's diagonal argument

I recently found Cantor's diagonal argument in Wikipedia, which is a really neat proof that some infinities are bigger than others (mind blown!). But then I realized this leads to an apparent paradox ...
0
votes
0answers
15 views

Average of left and right limits | Signum function, Heaviside step function, and Grandi's series

This question probably already has an answer but usually involves stuff that's way over the top of my head so I'm hoping for a simple explanation. In Adams, R. A., & Essex, C. (7th edition) ...
2
votes
2answers
133 views

A peculiar observation about infinity.

Let ${\sqrt2^\sqrt2}^{\sqrt2^...}=y$. Then $\sqrt 2^y=y$ $\implies \sqrt 2=y^{1/y}$ $\implies \sqrt 2 =1$ $\implies 2 =1$ !! but how come that be. Can anyone explain this and point out what is ...
0
votes
1answer
46 views

What is the formal definition of a limit at infinity?

I keep coming across two different kinds of answers to this question. The first definition: We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied: for every number $\...
6
votes
4answers
2k views

Circle revolutions rolling around another circle

I just watched this video, and I'm a bit perplexed. Problem: ...
3
votes
1answer
30 views

How do I calculate this limit when two terms tend to infinity at similar rates

In a particular problem that I am currently trying to solve, I have the following expression (this is not the entire expression, I have included only the terms involving $a_1$ and $b_1$), $\lim_{(...
3
votes
1answer
32 views

Interval notation: infinity, -infinity in closed interval

I was watching a video stream a little bit ago and noticed on an equation without context that had the interval $\left[{-\infty, \infty}\right]$. This was preculiar to me as I've never seen the ...
1
vote
1answer
56 views

precise definition of a limit at infinity, application for limit at sin(x)

(a) Write down the first principles definition of the statement $\lim\limits_{x→∞} f(x) = L$. For this I have that for every $ε >0$, there is a corresponding number $N$, such that if $N>0$, ...
0
votes
2answers
3k views

Must an infinite intersection of infinite sets be infinite?

If $A_2$ is a subset of $A_1$, $A_3$ is a subset of $A_2$, and this goes on infinitely and all contain an infinite number of elements, then is the intersection from $n=1$ to infinity, infinite as well?...
1
vote
2answers
71 views

did i use infinite wrong?

This algebra question is in Dutch and the original file van be found here: Question 19 Ill try to translate the important info needed to answer this question. $$s= \frac{(a+b)} { (ab)}$$ S= dpt ...
2
votes
3answers
38 views

Division of segments into infinitely many parts.

Let AB and CD be two segments, so that the length of AB is 1, and the length of CD is 2. If we divide AB and CD in infinitely many parts, how "long" would those parts be? I'm particularly interested ...