Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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Solve L'Hopitals problem

$$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x}{{\sec^2 3x}} $$ I used LH: $$\lim_{x\rightarrow \frac{\pi}{2}} \frac{\sec x \tan x}{6\sec 3x \sec 3x \tan 3x}$$ then: $$\lim_{x\rightarrow ...
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1answer
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Use L'Hopital's with this problem?

The problem is: $$\lim_{x\rightarrow 0^+} \left(\frac{1}{x}\right)^{\sin x}$$ I know the answer is $1$ because I checked with my graphing calculator, but how exactly do I get there? I got this far: ...
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2answers
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$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$

So the question is: $$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$$ First of all, I know we have to use Lhopital's rule. However, I just don't know how. Second of all, I thought in the end we ...
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0answers
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The sum of the infinite series [duplicate]

The sum of the infinite series: $$ \frac{1}{2} +\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +\cdots$$ I am able to find ...
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51 views

I need advice on calculating this limit of a function resulting in $-\frac14\pi$

I am looking for advice on solving this limit of a function. I am struggling to find the correct process: $$\lim\limits_{x\to-\infty}\operatorname{arccotg}\frac{x}{(x^2-4)^{\frac12}}$$
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2answers
32 views

What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
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5answers
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dirac delta integral with $\delta(\infty) \cdot e^{\infty}$

I have a question about this integral $ \displaystyle \int_{-\infty}^{+\infty} \delta'(x-3)e^{x^2}dx $ by integration by parts I get; $ \displaystyle ...
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100 views

Non-standard numbers and exponential form of Zeta function

Basic idea For a long time I was looking for a numerical system that would allow to compare infinite sets. In contrast to Cantor's approach that empathizes the possibility of on-to-one correspondence ...
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4answers
47 views

Why Zero divided by Zero is undefined and not Infinity [duplicate]

apologize in advance if this is a duplicate, but I found a lot questions related to this but none answering this specific question. My logic is: let's consider division the opposite of ...
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0answers
39 views

Good papers on Cantor's infinities?

I'm trying to get a mathematical treatment on Cantor's infinities, but can't seem to find papers merely by googling. Any recommendations?
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1answer
55 views

An infinite dictionary: countably infinite or uncountably infinite?

This question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The ...
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1answer
30 views

Comparing Infinities

Is it ever possible to say that $\infty = \infty$? For example, does the number of odd numbers ($\infty$) equal the number of even numbers ($\infty$)? Does does the number of odd numbers ($\infty$) ...
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2answers
25 views

Proving existence of numbers with intermediate value theorem

How do you use the intermediate value theorem to prove the existence of numbers? For example, with $f(x) = c^2 = 2$, how can I prove that $\sqrt2$ or a positive number $"c"$ such that $f(x)$ is true ...
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1answer
19 views

The Limit of an Integral Containing Exponentials

I am unsure how to show this. Suppose $\delta(s)$ defined on $(-\infty , s_*)$ is increasing and satisfies $\lim _{s\rightarrow s_*} \delta = \lim _{s\rightarrow s_*} \frac{d \delta}{d s} = \infty$ ...
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1answer
53 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
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124 views

Why the number of all reals is $2^{\aleph_0}$ and not ${\aleph_0}^{\aleph_0}$? [closed]

Here is a small intuition why it should be the later. Let $\omega$ be the number of all natural numbers. Then what is the smallest real number? We can write reals in binary form. Usual logic would ...
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Can a sequence have infinitely many limits among its subsequences?

Suppose we have an extended real (countably) infinite sequence $(x_n)$. Then consider all of its possible subsequences $(x_{n_k})$. We could then consider the set $$A = \{a\in ...
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26 views

Is there a relationship between the unbounded infinities and uncountable infinities? [duplicate]

When a function f increases without bound we say $f(x)=\infty$. How does this idea relate to, if at all with the infinite sets we study in set theory? To give a better understanding of why I'm ...
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2answers
78 views

Double Summation Trick

I have seen a couple of times the trick where $\displaystyle\sum_{i=1}^\infty \sum_{j=i}^\infty f(j)$ becomes $\displaystyle\sum_{j=1}^\infty \sum_{i=1}^j f(j)$ How does this work? I am so confused. ...
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1answer
35 views

Logic question requiring axiom of choice

Predicting Real Numbers Regarding the above question, the solutions require creating classes of sequences with representative sequences. How are those sequences constructed? How is it possible to ...
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722 views

Why do we distinguish between infinite cardinalities but not between infinite values?

More specifically, why are we "allowed" to denote $|\mathbb{N}|<|\mathbb{R}|$ but not $\sum\limits_{n\in\mathbb{N}}1<\sum\limits_{r\in\mathbb{R}}1$? Can we distinguish between "countable ...
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2answers
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Alternate Axiom of Infinity

The Axiom of Infinity states that there is a set $S$ containing $\varnothing$ such that if $x$ is an element of $S$ then so is $x\cup\{x\}$. Is the following variant equivalent? There exists a ...
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205 views

Two similar method to calculate one equation get different answer

Method1:$$\lim_{x\rightarrow0}({\frac{e^x+xe^x}{e^x-1}}-\frac1x)=\lim_{x\rightarrow0}({\frac{e^x+xe^x}{x}}-\frac1x)=\lim_{x\rightarrow0}(\frac{e^x+xe^x-1}{x})=\lim_{x\rightarrow0}(2e^x+xe^x)=2$$ ...
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0answers
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Is there a change-of-variables solution for integrals from negative infinity to a constant?

I found a fantastic and generalizable substitution technique for computing definite integrals that go to infinity from either negative infinity or a constant, regardless of the function (sorry for the ...
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2answers
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Is $\frac00=\infty$? And what is $\frac10$? Are they same? Does it hold true for any constant $a$ in $\frac{a}0$ [duplicate]

I know that $\lim_{x\to0}\frac{x}{x}=$ 1. But in my text book, it is written that it is $\infty$ and even $\frac10=\infty$. But how is it possible? And are they both same? What is the difference ...
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2answers
48 views

What is the value of $0\times \infty$? (in $[0, +\infty]$)

In $[0,+\infty)$, $0^+\times +\infty$ can be any number in $(0,+\infty)$ so is undetermined; (in which $0^+$ means when a variable approaches to $0$). Because $\lim_{x\rightarrow ...
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1answer
98 views

Is 1 + 1 + 1 … a finite number? [closed]

Just a simple question whether the "number" $x = 1 + 1 + 1 ...$ is a finite number. On one hand, you can think of x as the result of a never-ending process; $0$ is finite and $n + 1$ is finite when ...
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4answers
146 views

Can a number have an uncountably infinite amount of digits?

I'm an extreme mathematical layman, so please excuse the probable ignorance and awkward phrasing of this question. Is there such thing as a kind of number which has an uncountably infinite amount of ...
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1answer
65 views

What is ordinal expression of $\infty$? [closed]

$\infty$ - is cardinal expression? ?Origin of a line ray is ordinal expression of $\infty$, if distance from the $\infty$ to the $0$ origin of line ray?
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7answers
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One divided by infinity is not zero? [duplicate]

I know that $\frac{1}{\infty}$ is undefined. But my question is - can we say that $\frac{1}{\infty}\neq0$ ? I've got some idea how to explain that: Let's say we have a random-number generator that ...
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Does 1/∞ = 0.0̅01? Can some tell why not? [closed]

im just an 8th grade math student going into geomotry 1, but im curious to know if 1/∞ = 0.0̅01 (or 0.000...1). If that is so, then because 0.9̅9=1, then 0.0̅01 would be 0, thus proving that 1/0=∞ its ...
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2answers
59 views

Third axiom of Kolmogorov axioms

Let us define for a countably infinite set $S$ of real numbers that can be enumerated as $x_1,x_2,\cdots$, $$P(S) = \sum_{x \in S}p(x) = \sum_{i=1}^\infty p(x_i) = \lim_{n \to \infty}\sum_{i=1}^n ...
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81 views

How to find the limit of a sequence?

Question: If $0 < x < \frac{\pi}{2}$ and $f_k(x) = \tan(x)+\frac{1}{2}\tan(x/2)+ ...+\frac{1}{2^k}\tan(x/2^k)$. In Sigma Notation: $$f_k(x) = \sum_{n=0}^k \frac{1}{2^n}\tan\frac{x}{2^n}$$ ...
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93 views

$\ln{\left(\frac{1}{0}\right)} = -\infty$?

I have shown it using a theorem that I made, but I am not sure, as $\lim_{\alpha \to 0^{-}}{\left(\frac{1}{\alpha}\right)} = -\infty$, and $\lim_{\alpha \to 0^{+}}{\left(\frac{1}{\alpha}\right)} = ...
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858 views

W. Mückenheim claims a severe inconsistency of transfinite set theory; true? [closed]

The abstract for a paper on arxiv.org (http://arxiv.org/pdf/math/0408089v3.pdf) reads (with my emphasis): "Transfinite set theory including the axiom of choice supplies the following basic theorems: ...
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166 views

Is infinity an element of the natural numbers? [duplicate]

I am wondering, whether $\infty \in \mathbb{N}$.
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Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
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1answer
29 views

Group action of $GL(2, F)$ on the projective line $P(F)$

I refer to section 8.3, page 119 of Algebra, A Computational Introduction by John Scherk. It is about group action of $GL(2, F)$ on the projective line $P(F) = F \cup \{\infty\}$. Given a matrix ...
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2answers
59 views

Calculating $\lim_{(n,x) \rightarrow (\infty, 1^+)}x^n$?

Obviously, when $|x| \leq 1$, $\lim_{n \rightarrow \infty}x^n$ has a value such that $|x^n| \leq 1$, and $|x^n| \rightarrow \infty$ for $|x|>1$. However, this led me to wonder, what would happen ...
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1answer
32 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
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1answer
116 views

Can we formally distinguish between actual and potential infinities?

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the ...
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219 views

Circle revolutions rolling around another circle

I just watched this video, and I'm a bit perplexed. Problem: ...
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132 views

Does $\tan (x)$ equal $\frac{-1}{x-\frac{\pi}{2}}+\frac{-1}{x+\frac{\pi}{2}}+\frac{-1}{x-\frac{3\pi}{2}}+\frac{-1}{x+\frac{3\pi}{2}}+…$?

I set my Year 12 students a question involving the sums of rational functions $\frac{1}{x-n}$. The graph of a sum of these functions looks an awful lot like a tan graph. This led me to ask: Does ...
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1answer
107 views

Infinity xkcd style: can a turing machine exist?

I recently read this xkcd comic. It's about a guy who simulates a universe by a Turing machine (specifically, Rule 101, a cellular automaton), by laying down infinite rows of rocks, each row ...
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1answer
45 views

The real projective line and $1/\infty$

so I came up with this idea: the real projective line defines that $\infty = - \infty$. What if I divide any value $x$ (not equal to $\infty$) by infinity? Would that be 0? or "something" between ...
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7answers
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After switching a lamp on and off infinitely many times in one minute, is it on or off? [duplicate]

So we have a lamp. It's switched on. let's represent its state of being switched on with associating it with $1$ and being off with $-1$. after half a minute passes, you turn it off, after another ...
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Possible proof for the multiple sizes of infinity [closed]

One can easily relate all regular polygons to the triangle: triangle = 1 triangle square = 2 triangles pentagon = 3 triangles hexagon = 4 triangles and so on and so forth… A circle is basically ...
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137 views

What we get if we add 1/2 infinite times [closed]

I want to know if this is correct We have this sums: $$S1=1-1+1-1+1-1+1-1+1-1...=\frac12$$ $$S2=1-2+3-4+5-6+7-8...=\frac14$$ $$S3=1+2+3+4+5+6+7+8...=-\frac{1}{12}$$ If we take ...
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7answers
154 views

The meaning of the symbol $\infty$ in Spivak's calculus book

Spivak in "Calculus" writes ... symbols of $\infty$ and $- \infty$ are purely suggestive: there is no number $``\infty"$ which satisfies $\infty \geq a$ for all numbers $a$. What is the meaning ...
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2answers
64 views

I am trying to find the limit of P(x)

When I am looking for a $\lim\limits_{x \to -1} P(x)$ where P(x)$= \sum \limits_{n=1}^\infty \left( \arctan \frac{1}{\sqrt{n+1}} - \arctan \frac{1}{\sqrt{n+x}}\right) $ do I have to ignore a ...