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

learn more… | top users | synonyms

-1
votes
2answers
423 views

What digits is the “number” infinity composed of?

I have seen from past posts on the topic of infinity that there is some ambiguity with the concept infinity and whether it is a number etc. From what I can gather the terms number and infinity are ...
1
vote
1answer
39 views

Limits to infinity (n)

Hi I have a question regarding finding the values of limit for the following equation. The question states to find the following limits: $$ \lim_{x\to\infty}\left(\frac ...
3
votes
6answers
348 views

What is larger: The inside or the outside of the infinite circle? [closed]

Assume a circle with radius $R$ in a plane. Let $R$ go to infinity. What is larger: The inside or the outside of the circle? EDIT My naive way of thinking about "largeness" was just to compare ...
16
votes
3answers
3k views

Is an infinite line the same thing as an infinite circle?

Imagine that you are sitting next to a line that extends infinitely in both directions. Is it possible to distinguish it from an infinite circle? From my poor understanding of topology, I would ...
1
vote
5answers
209 views

Limits to infinity Finding Constant Number

Hi I have a question regarding of limits to infinity please help which I need to find the constant number for a and b. Please help! Thank You! The question states the user to find the following ...
1
vote
2answers
53 views

Find the Limit (n to infinity)

Hi I have a question regarding of limits to infinity please help! Thank You! The question states the user to find the following limit: $ \lim_{n\to\infty} n^2 ({\sqrt[n]{x}-\sqrt[n+1]{x}}) $ ...
2
votes
0answers
36 views

Countability of unions versus products

Let $D_{n}$ be a set with $2^{n}$ elements for $n=1,2,...$. Let $A = \bigcup_{n=1}^{\infty}D_{n}$, and let $B = \prod_{n=1}^{\infty}\{0,1\}$. Let $A_{k} = \bigcup_{n=1}^{k} D_{n}$, and let $B_{k} = ...
2
votes
1answer
76 views

Is this improper integral answer correct?

So I'm working on improper integrals and con/divergence and want some assurance that I've done the following correctly. $\int^∞_{-∞}cos(\pi t)$ As far as I'm aware this is convergent if and only if ...
4
votes
6answers
279 views

justification of a limit

I encountered something interesting when trying to differentiate $F(x) = c$. Consider: $\lim_{x→0}\frac0x$. I understand that for any $x$, no matter how incredibly small, we will have $0$ as the ...
6
votes
4answers
554 views

What happens if I toss a coin with decreasing probability to get a head?

Yesterday night, while I was trying to sleep, I found myself stuck with a simple statistics problem. Let's imagine we have a "magical coin", which is completely identical to a normal coin but for a ...
-1
votes
1answer
81 views

Depth of infinite direct sum

Let $R$ is a local ring, from the depth lemma, we can get $\operatorname{depth}(R\oplus\dotsb\oplus R)=\operatorname{depth}(R)$, here the direct sum is finite, how about the infinite case? By the ...
1
vote
1answer
141 views

In an infinity of choices, is it possible to guess the correct one?

So I've been thinking about the infinite universes model, where each possible action or event creates a new universe for each outcome. For example, if you flip a coin there will be one universe in ...
0
votes
3answers
139 views

Epsilon-Delta Proof at infinity

Let $n \in \mathbb{N}$ and $y \in \mathbb{R}$ and $0<y<1$. Let also be $f(y)=y^n$ and $g(y)=y^{n+1}$. $$ \lim_{n \to \infty} \cfrac{f(y)}{g(y)} = L $$ What is the value of $L$ using the ...
0
votes
4answers
288 views

Is $\infty / \infty = 1$?

Lately, my friend and I were arguing about what $\infty / \infty$ equals. My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as ...
1
vote
2answers
246 views

Indeterminate form limits question

$$\lim_{x\to 0}\frac{10^x - 2^x - 5 ^ x + 1 } {x\tan x} $$ This is an indeterminate limit. I want help in solving this problem. Thanks in advance
0
votes
1answer
182 views

Countable and Uncountable sets

Is $\mathbb{N}\cup\{a\}$, for some $a\not\in\mathbb{N}$ countable or uncountable? $\mathbf{Attempt: }$ It is true that a set is countable if there exists an injective function $f : S → N$ from $S$ to ...
2
votes
1answer
116 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
1
vote
2answers
120 views

Limit when an expoent goes to infinity

Please could someone help me and see if my solutions are correct for these two limits Let $n \in \mathbb{N}$ and $y \in \mathbb{R}$ and $y>0$. Case 1 $$\lim_{y \to \infty} ...
0
votes
1answer
39 views

Number of k-permutations that have odd number of an element

I want to find a recurrence relation $h_k$ for the number of k-permutations of $\{\infty a,\infty b, \infty c, \infty d \}$ that have an odd number of a's. I let $h_0=0$ because there is no odd ...
39
votes
7answers
3k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
2
votes
3answers
2k views

Why is the integral of sec^2(x) from 0 to pi infinity?

Why is it, if you take the integral of sec^2(x) from 0 to pi, my calculator returns "infinity" as the answer, but according to the second fundamental theorem of calculus, I got 0 with my own work. I ...
0
votes
2answers
295 views

Is this true that if limit approaches infinity the function equals to zero?

I would like to know if the following is true. If $$\lim_{z\to \infty} 1/f(z) = \infty$$ is that equivalent to $$\lim_{z\to \infty} f(z) = 0?$$
0
votes
2answers
85 views

Questions about hyperbolas and integration

I have a couple of questions regarding hyperbolas and their integrals. If it's too much, don't feel like you have to answer all 3 questions. My first question: The integral of a function like 1/x^2 ...
8
votes
10answers
3k views

Smallest next real number after an integer

This might be a silly question, but is it possible at all for n.00000...[infinite zeros]...1 to be the next real number after n? If not, why not? Firstly, I know (I think) that $$\lim_{x\to \infty} ...
-1
votes
4answers
152 views

Does $\infty^0=1$?

I was wondering if $\infty^0=1$. Some people have told me that there is no answer; it is undefined. Others have told me that the answer is $1$, using the rule $a^0=1, \ a\neq 0$. If it is truly ...
0
votes
2answers
45 views

Evaluating a limit as $x\to -\infty$

I am trying to evaluate $$ \lim_{x \to -\infty} \left(1+ \frac{1}{x}\right)^{x²}. $$ I'd say it tends to 0, 1 or something linked to $e$ but I have no clue how to prove this... I'm getting really ...
0
votes
3answers
39 views

Computing the limit of this function

So I have an improper integral: $$ \int_0^\infty \frac{13x}{x^2+1}-\frac{65}{5x+1} dx $$ I have solved the integral into this: $$ \lim_{t \to \infty} ...
14
votes
5answers
2k views

Is half a pie as big as a whole pie?

I am reading an e-book called To Infinity and Beyond by Dr. Kent A Bessey. In the book the author makes the claim that Georg Cantor made a discovery "where half of a pie is as large as the whole". In ...
1
vote
1answer
17k views

What is zero times infinity? [duplicate]

If any number times zero is zero and any number time infinity is infinity, then what do you get when you multiply zero times infinity? Do they cancel one another out and equal any number since any ...
0
votes
2answers
87 views

How to show by the Root Test that $\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$ converges or diverges

How do I show by the Root Test that $$\sum\limits_{i=1}^\infty (2n^{1/n}+1)^n$$ converges or diverges? This is what I have done so far. Since we take $\sum\limits_{i=1}^\infty \sqrt[n]{|a_n|}$, we ...
0
votes
2answers
57 views

What this statement is really saying to prove one Real number has missed the bijection with Integers?

In a Combinatorics text, I find this: Not all infinite sets have the same cardinality. Consider the set of all integers and the set of all reals. Assume that the set of reals can be put in ...
1
vote
1answer
2k views

Applications of infinity in real life [duplicate]

I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does ...
1
vote
1answer
203 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
-2
votes
1answer
161 views

infinity between two points on a line

I remember from school that the number of points on a section of a line is infinite. On the other hand, when you reach the number two in a number sequence, that is a number and how big the number is, ...
8
votes
4answers
2k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements (including subatomic particles), so ...
0
votes
3answers
142 views

Interpretations of $\frac{\infty}{\infty}$

I am trying to understand the physical sense of the mathematical construct $\frac{\infty}{\infty}$ Suppose we have a function $f(x)$ representing some physical construct depending on a "quantity" $x$ ...
2
votes
5answers
449 views

Limit of $(n-k)! \cdot n^k$ as $n$ approaches infinity

Is it true that $(n-k)! \cdot n^k$ tends to $n!$ as $n \to \infty$? I think it is correct but can't think of a satisfying proof.
0
votes
2answers
1k views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
3
votes
0answers
205 views

Limits of infinite processes that terminate in finite time - checking my understanding?

I am a computer scientist by training, but have a fair amount of math background that I've picked up through classes, teaching, and general interest. A student of mine posed a question to me. I think ...
1
vote
1answer
82 views

A function that escapes to infinity with a finite input

I was wondering whether there exists a function that escapes to infinity with a finite input. For a specific example, how about $f(0)=0$ and as $x$ tends to $10$, $f(x)$ tends to infinity. The use of ...
0
votes
2answers
86 views

Evaluating a limit with infinity

I'm taking the limit as x approaches infinity from the left (-) of: $$ \sqrt{x^2+2x}- \sqrt{x^2-2x} $$ However I'm not sure how to go about this. I'm at: $$ \sqrt{ \frac{x^3+4x^2}{x+2x}}- \sqrt ...
2
votes
2answers
73 views

Finding asymptotes of $(2-x^2)/(e^x)$

I was trying to solve some exam question on calculus 1, and i found this "Sketch the graph of $(2-x^2)/(e^x)$" I'm interested to find Horizontal Asymptotes of the graph. 1) when x approaches ...
0
votes
2answers
105 views

Can real number infinity be bigger than other real number infinity?

I know that 2 countable infinities are considered equal because you can pair each element in one set two an element in another one. But, for example, if we let all real numbers between 3 and 5 be ...
1
vote
2answers
432 views

If the Kleene star of countable sets is countable, how are the real numbers uncountable?

The formal languages we use to represent number systems are interchangeable, which is why we don't hesitate to use different notations, e.g. hexadecimal, octal, binary, etc... to represent the reals. ...
6
votes
3answers
774 views

Limits at infinity

I'm working with limits at infinity and stumbled upon this exercise where I want to evaluate the indicated limit: $$\lim_{x \to \infty} \frac{1}{\sqrt{x^2-2x}-x}$$ I tried to solve it by doing the ...
11
votes
4answers
529 views

Partitioning the naturals into an infinite number of large sets

Is it possible to partition the positive integers into an infinite number of disjoint large sets ?
1
vote
7answers
243 views

Does infinite equal infinite?

I have a question. Let $x$ be infinite. $$2x=\infty\times2, \quad 2x=\infty$$ So actually, does $2x=x$?
1
vote
1answer
151 views

Function with a constant infinite order derivative, infinite final value, 0 initial value, and graph that resembles geometric growth

Please forgive my vocabulary & usage because I'm only a math amateur, so I'll try to describe this the best I can. Does such a function exist that has an infinite order derivative with a constant ...
1
vote
3answers
251 views

Why are irrational numbers uncountable and rationals contable?

Question 1: Why are irrational numbers uncountable and rationals contable? I really struggle to understand this. I initially thought it had something to with the fact that between any two numbers ...
0
votes
1answer
30 views

Probability on the plane

Problem. On the Cartesian plane with origin O and x- y-axes, I randomly pick a point P. What is the probability that the line segment OP has a slope at least 1? Is the answer 1/4 or 1/2? answer = ...