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

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2answers
76 views

the relationship between $f(x)$ and $dx$ in $\int_a^b f(x)\,\mathrm{d}x$

In this example, In the last equation of this example, obviously, the two $dx$ are canceled in the left before we get the right, this indicates that the relationship between $f(x)$ and $dx$ ...
0
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3answers
61 views

How does differentiation work?

I am a physics student and my teacher told me, to find the instantaneous velocity of an object, reduce the time interval to a very small extent. May the time interval be very very very close to 0, ...
35
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20answers
2k views

Problem with basic definition of a tangent line.

I have just started studying calculus for the first time, and here I see something called a tangent. They say, a tangent is a line that cuts a curve at exactly one point. But there are a lot of lines ...
0
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2answers
64 views

What is the value of $\tan (\frac{\pi}{2} - \epsilon)$? [closed]

$$\tan(\frac{\pi}{2}-\epsilon)$$ By epsilon I mean an infinitesimal.
0
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2answers
73 views

Cauchy's contribution

Sometime, I believe perhaps 2 years, ago I asked a question about breakthroughs, such as those within mathematics and physics which may lead a whole discipline forwards in many ways. One example from ...
7
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6answers
637 views

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
1
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1answer
51 views

Proving a Function Continuous with Non-Standard Analysis

I am reading a text on non-standard analysis. I need to prove the following: Suppose that $f$ is non-decreasing on the real interval $[a,b]$ and that $f$ satisfies the intermediate value property. ...
0
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5answers
368 views

Why can't the reals be constructed from the infinitesimal?

If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then Why can't the reals, which demands, simply, unlimited precision (this ...
3
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1answer
77 views

Do we have $\int f dxdy = \int fdydx$ or $\int f dxdy = -\int f dydx$?

If $f : \mathbb R^2 \to \mathbb R$ is an integrable function, then do we have $$ \int f dxdy = \int f dydx $$ or $$ \int f dxdy = -\int f dydx? $$ (I am leaving the domain of integration as it does ...
0
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2answers
72 views

proof that $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = \infty $

I started learning infinitesimally math and I have the following question: Is the following sentence true $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = 1 $ I can see that it tends to ...
1
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10answers
951 views

What's an example of an infinitesimal?

If you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give them? What I am asking for are specific techniques for explaining infinitesimals to students, ...
5
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2answers
254 views

How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...
0
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2answers
821 views

$1/\infty$ is zero or infinitesimal? [closed]

Since $\infty>0$, so $1/\infty >0$, thus I think $1/\infty$ should be infinitesimal, but the calculus book says $$\lim_{x \to \infty} \frac{1}{x}= 0$$ So is $1/\infty$ zero or infinitesimal ? ...
6
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3answers
193 views

Are there concepts in nonstandard analysis that are useful for an introductory calculus student to know?

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this ...
7
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1answer
586 views

Infinitesimal calculus

I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ...
3
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1answer
76 views

Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
4
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3answers
663 views

Why don't infinite sums make any sense?

Using the infinite sum series, an infinite sum of (1/5)to the nth power, where n goes from zero to infinity, the general summation equation tells us that the answer is 5/4. However, how is this ...
3
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2answers
82 views

Definite Integral of a infinitesimal

I did not study math, but have some foundations in it. I have been looking through some books on nonstandard analysis, and have (what I consider to be) a pretty simple question which I haven't been ...
0
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0answers
9 views

Link to question regarding treating differential operator as a ratio [duplicate]

I have attempted to find the post which provides an explanation as to the circumstances in which we can treat $\frac{dy}{dx}$ as a ratio which appears to be used in solving separable DE's, but I have ...
2
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2answers
48 views

Ceiling function of an infinitesimal

I was working with infinitesimals and I came across the problem: what is the ceiling function of an infinitesimal? Wolfram Alpha says an infinitesimal equals zero, so therefore the ceiling function of ...
2
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4answers
557 views

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the ...
0
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1answer
16 views

Make $f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$ be the infinitesimal of the highest order

Here is the question: Find $a$ and $b$, letting $$f(x)=\sin x-\frac{x+ax^3}{1+bx^2}$$ be the infinitesimal of the highest order when $x \to 0$, and find that order. According to the key, ...
6
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8answers
228 views

Products of Infinitesimals

In my physics class my professor was abusing the derivative, as per so many physics classes I've been in. This time, he took the quantity $(x+dx)(y+dy)$ and argued that the $dxdy$ term should ...
37
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15answers
4k views

Is it a new type of induction? (Infinitesimal induction) Is this even true?

Suppose we want to prove Euler's Formula with induction for all positive real numbers. At first this seems baffling, but an idea struck my mind today. Prove: $$e^{ix}=\cos x+i\sin x \ \ \ ...
16
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3answers
2k views

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 ...
5
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1answer
76 views

Interpretation of $d\phi(z)$ in differential geometry

In "Exercises and Solutions in Mathematics", Ta-Tsien, 2nd Edition, exercise 3343. Statement of the exercise Let $(\mathbb{H}, g)$ be the two-dimensional hyperbolic space, where \begin{equation} ...
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1answer
29 views

Dual number division

I was reading the Wikipedia article on dual numbers and I read the first part. Then on my own I tried to find how to divide two dual numbers. I came up with: ...
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1answer
495 views

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 ...
2
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1answer
32 views

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 ...
3
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4answers
83 views

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 ...
0
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0answers
28 views

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 ...
14
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8answers
467 views

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 ...
1
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2answers
93 views

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 ...
2
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1answer
103 views

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 ...
0
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2answers
25 views

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 ...
1
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1answer
40 views

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 ...
1
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1answer
50 views

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 ...
1
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0answers
28 views

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. ...
0
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1answer
28 views

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 ...
1
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1answer
61 views

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 ...
3
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2answers
60 views

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 ...
5
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1answer
69 views

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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2answers
91 views

Historical Approach to $\lim_{x \to 0} \frac{e^{\alpha x} - e^{\beta x}}{x}$, without L'Hospital's Rule

I encountered this problem, amongst others, in the slightly older Calculus textbook Piskunov's Differential and Integral Calculus when I was working with a student: Calculate the limit $$ \lim_{x \to ...
4
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2answers
65 views

Basic question about nonstandard derivative

I'm trying to understand how the nonstandard derivative works. For instance, consider the function $f(x) = \frac{1}{2} x^2$ The derivative is $f'(x) = st \left( \frac{\frac{1}{2}(x + \epsilon)^2 - ...
2
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6answers
112 views

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 ...
18
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1answer
671 views

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 ...
0
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0answers
53 views

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 $C(\mathbb{R})$ ?
2
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2answers
77 views

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 ...
0
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1answer
67 views

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. :)
4
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0answers
49 views

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 ...