# Tagged Questions

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

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### Could we assign a numerical value to an infinitesimal?

The common definition of 'infinitesimal' is that they are quantities too small to be measured or perceived. If we base the mathematically definition entirely on the common definition it yields a ...
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### Does this idea make sense? (infinitesmals and positive change)

I didn't know where else to ask this, so here goes... (caveat: I've been away from math for awhile) I'm trying to "say" symbolically that continued positive infinitesimal changes eventually (not at ...
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### Find $\lim_{x\to0}\frac1{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right)$ without L'Hopital's rule

Find the following limit $$\lim_{x\to0}\frac1{x^3}\left(\left(\frac{2+\cos x}{3}\right)^x-1\right)$$ without using L'Hopital's rule. I tried to solve this using fundamental limits such as ...
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### Combine probability and infinitesimals

If we pick a real number from $[0, 1]$ (with uniform probability measure), this number had probability zero to be picked. Can we canonically assign an infinitesimal $\epsilon$ to the event $A_x$ that ...
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### Intuition for a physical real line vs. a physical “hyperreal line”

As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach". "We have no way of knowing ...
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### Is there such a thing as “hypertopology” (analogous to the hyperreals)?

The hyperreal number system adds infinities and infinitesimals, allowing Calculus to be done using these things instead of limits (sort of like when calculus was originally invented, but with ...
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### Mapping $\mathbb{Z}$ onto $\mathbb{Q}$

There is a problem that calls for the mapping the set of all the positive integers to the set of all positive rational numbers in order to prove they have the same cardinality. I know more the common ...
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### Find Maclaurin expansion of $y=2^x$ to $x^4$

Find Maclaurin expansion of $$y=2^x\text{ to } x^4$$ This is my try. We have $\displaystyle 2^x=e^{x\ln 2} =\left[1+\frac{x^2}2+\frac{x^3}6+\frac{x^4}{24}+o(x^4)\right]^{\ln 2}$ with $o(x^4)$ is ...
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### Infinitesimal generator as a derivation in $SO_2$

Suppose we are looking at $SO_2(\mathbb{R})$. The infitismal generator can eb found using Taylor approximation in form of a matrix, $$X=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.$$ However I saw ...
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### Does probability theory require infinitesimals to work?

I'm taking an intro to probability class, and my book lists this as an axiom: $$P(\bigcup_{k=1}^{\infty}A_k) = \sum_{k=1}^{\infty}P(A_k)$$ where $A_k$ is an event and $P(A_k)$ is the probability of ...
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### Infinitesimal Transformation. Importance of Infinitesimal Transformation

My teacher ask me to start to study about Hamilton systems, Noether theorem. I am not advanced in that kind of study, so the teacher want from me to see some easy research, some proofs, step by step. ...
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### Is there any relationship between Gödel numbers associated with proofs of undecidable theorems and infinitesimals?

I recall from my reading of the popular book, Gödel, Escher, Bach some years ago that Hofstadter speculated on the value of Gödel numbers of undecidable theorems. If I recall correctly, he suggested ...
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### Non-standard analysis - infinitesimals and archimedean property

I got a question about infinitesimals in non-standard analysis. If I understand correctly, they are defined to be the number that is closest to zero. However, at the same time, they satisfy all the ...
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### What happens if to introduce infinite and infinitesimel quantities this way?

What if to introduce $\varepsilon$ and $\omega=1/\varepsilon$ from the following equation as a definition? $$\left(1+\varepsilon\right)^{1/\varepsilon}=e$$ or such $\varepsilon$ that ...
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### Can this sequence be a hyperreal number? What would be its real part?

Consider the sequence $\{a_n\} = \{\sin(n) \mid n\in \mathbb N \}$. Can this sequence be viewed as a hyperreal number? What could be its real part? Any intuition would be highly appreciated :) ...
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### More numbers between $[0,1]$ or $[1,\infty)$?

There are infinitely many real numbers between any two real numbers, therefore there are infinitely many real numbers in the range $[0,1]$ as there are in $[1, \infty)$. In a mathematical sense, are ...
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### Why is zero the only infinitesimal real number?

I am currently reading Elementary Calculus: An Infinitesimal Approach by H. Jerome Keisler and was wondering if someone could help me with an aspect treated in the book. On page 24 he says a ...
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### Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
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### Are Gaussians a basis for the vector space of continuous functions?

How can I prove (or disprove) that the Gaussian function family: $f_{\mu,\sigma}(x)=e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$ Are a basis for $C(\mathbb{R})$ ?
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### Convergence of $\int\frac{\arctan x}{x} dx$

I can't find function to bound this integral in the intervals from $1$ to $+\infty$, to prove if it converges. $$\int _1^{\infty }\frac{\arctan x}{x}dx$$ Any idea? How can I refute this if it is ...
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### Convergent of integral of $1/x^x$ [closed]

I need to prove if this integral converges (on interval $[1,+\infty))$: $$\int _1^{\infty }\frac{1}{x^x}\;dx$$ Has anybody any idea how to do it? thank you. :)
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### Rigorous Justification of Infinitesimal Techniques

As you may know that there are a bunch of heuristic techniques in physics to make integrals converge. For example, when we define a following Fourier transform, we add a positive infinitesimal and let ...
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### What is the topology of the hyperreal line?

Denote by $\Bbb R$ the real line and by $\Bbb R^*$ the hyperreal line. For any real numbers $x < y < z$ and infinitesimal $\epsilon$ the following holds: \forall a,b,c \in \Bbb ...
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### infinitesimal intervals in physics

The density of states of a system in an interval $[E, E+dE]$ is given implicity by $dV = D(E)dE$ (Or I suppose explicitly, by $D(E) = \frac {dV}{dE}$, but we'll be integrating it anyway, so it doesn't ...
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### Size of a geometric point

It is well known that the geometric points do not have any length, area, volume, or any other dimensional attribute, also geometric object (for example "line") is made up of a infinite number of ...
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### Does it make sense to ask if there are as many real numbers between 1 and 2 as between 0 and 1?

If yes, how does one go about finding the answer? Can the same question be asked for the hyperreals too?
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### Using differentials to optimize a function

I've read in a paper by Tevian Dray an alternative way to solve optimization problems manipulating "differentials". Here is an example of how it works (next I quote the paper). Consider the ...
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### hyperreals standard part inconsistency

$\def\st{\operatorname{st}}$ I'm studying non-standard calc from Keisler's book. Taking "standard part" rule doesn't make sense... its not commutative. e.g. $a$ is finite non infinitesimal $b,c$ ...
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### Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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### When “magnifying infinitesimals” why dont they have curvature ? (non standard) Infinitesimal calculus

Im reading https://www.math.wisc.edu/~keisler/calc.html. If you open up the chapter $2$ pdf... The $2$ diagrams (1st on page $14$ of the pdf (not the text book), 2nd on page $15$) have me confused. ...
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### What is the limit ${{\lim }_{x\to\infty}}x^\epsilon$ for an infinitesimal $\epsilon$?

What is the limit $${{\lim }_{x\to\infty}}x^\epsilon$$ for an infinitesimal $\epsilon$? Does it give zero or infinity? Note that I'm considering the infinitesimals described in ...
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### Is this a valid thing to do to this differential equation?

Don't tell me how to solve it, but I've been trying to use the following equation to get r which is distance) as a function of t. $$\frac{GMm}{r^2}= -ma$$ I've been working on it for about a year ...
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### Confused between infinitesimally small and $0$

Consider these two cases: 1) Let $I_n$ be the closed interval $[\frac{1}{n}, 1]$. Then $\bigcup_{n = 1}^{\infty} I_n = (0,1]$. 2) Take the two sets $y = e^x$ and $y = 0$ (the $x$-axis) in ...
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### Proving: if $|a|<\epsilon \forall \epsilon>0$ then $a=0$ using a direct proof

I am asked to prove: if $|a|<\epsilon,\forall \epsilon>0$, then $a=0$ I can prove this as follows. Assume $a \not= 0$ I want to show then that $|a| \geq \epsilon$ for some $\epsilon$ We ...
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### Method for computing polar coordinates surface element?

I have tried to compute the "classical" surface element in polar coordinates for volume integration (i.e. $dx\ dy=r \ dr\ d\theta$) through this method:  \left\{ \begin{array}{c} x=r \cos \theta\\ ...
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### Is $\textrm{d}x \in \mathbb{R}$?

Is an infinitesimal a real number? Can "abuse of Leibniz's notation" be justified by claiming that an infinitesimal is a real number? If not, what is an infinitesimal?
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### Is non-standard analysis worth learning?

As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides ...
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### Multivariable functions limits and paths

In order to approach a point as (0,0) there many directions to do so. A whole 360 degrees actually. So between [0,360) degrees there are actually infinite directions. My question is why does it ...
### Order of infinitesimal of $\frac{\cos x}{x}$b for $x \to \infty$
I can prove that $\frac{\cos x}{x}$ is an infinitesimal for $x \to \infty$ with the squeeze theorem. But trying to find the order of infinitesimal, I'm not sure if my reasoning is valid. Here's what ...