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That's interesting, thanks for sharing. The main question is not answered though: " How many terms should I grab to go safe for every case? Why doesn't it suffice to take just the 1st non-zero term? " Then they work with the limit: \[\lim_{x \to 0} \frac{tan(x) - sin(x)}{x^3} \] The answer is 1/2, using the limit calculator: https://www.symbolab.com/solver/limit-calculator/\lim_{x\to0}\frac{\left(tan\left(x\right) - sin\left(x\right)\right)}{x^{^3}} The limit calculator uses L'Hôpital three times and then plugs in the value 0. What I was asing in my first post, is it possible to plug in an infinitesimal value? Is it possible to calculate limits using infinitesimals?

Normally limits are used instead of infinitesimals, but is it possible to calculate limits using infinitesimals? For example: \[ \lim_{x \to 0} \frac{sin(x) - x}{x^3} \] this is usually solved by applying L'Hopital's rule 3 times and the answer is -1/6: https://www.symbolab.com/solver/limit-calculator/\lim_{x\to0}\frac{\left(sin\left(x\right) - x\right)}{x^{^3}}

Every point of a number line is assumed to correspond to a real number. https://en.wikipedia.org/wiki/Number_line Is it possible to find points corresponding to infinitesimals on a number line? I mean finding an infinitesimal between two neighbouring points (between two real numbers). I am assuming that every point is surrounded by neighbourhood. I got this idea of neighbouring points from John L . Bells' book A Primer of Infinitesimal Analyis (2008). On page 6, he mentions the concept of ‘infinitesimal neighbourhood of 0’. But I think he would not consider his infinitesimals as points because on page 3 he writes that "Since an infinitesimal in the sense just described is a part of the continuum from which it has been extracted, it follows that it cannot be a point: to emphasize this we shall call such infinitesimals nonpunctiform."

The limit defintion of derivative in my previous post contains only the symbols h (corresponding to Δx) and dx. There is no δx. It seems to me that the introduction of "differential calculus" gives rise to symbol δx. Then there seems to appear two representations: f'(x) = dy/dx f'(x) = δy/δx I think it is possible the usage of δy/δx was chosen to escape the problem arising in real number calculus, the problem of 0/0.

I am beginning to suspect that calculus is not based on real numbers. Look at the definition of the derivative: \[\frac{dy}{dx} = \lim_{h\to\ 0}\frac{f(x+h) - f(x)}{h}\] where h is finite. What is dy/dx? An infinitesimal ratio? A ratio of two infinitesimals dy and dx ? It seems to me that we are not dealing with real numbers anymore if dy and dx are not real numbers.

I don't know. Maybe the answer can be found in the book I am studying. John L. Bell is defining the ‘derivative’ of an arbitrary given function f : R → R. For fixed x in R, define the function g: Δ → R by g(ε) = f(x + ε) so that f(x + ε) = f(x) + εf'(x) is the fundamental equation of the differential calculus in S for arbitrary x in R and ε in Δ.( Δ may be considered an infinitesimal neighbourhood or microneigbourhood of 0). Also he is stating Microaffiness Axiom: For any map f:Δ → R there exist unique a, b ϵ R such that f(ε) = a + bε for all ε ϵ Δ He draws a conclusion: Our single most important underlying assumption will be: in S, all curves determined by functions from R to R satisfy the Principle of Microstraightness. The Principle of Microaffineness may be construed as asserting that, in S, the microneighbourhood Δ can be subjected only to translations and rotations, i.e. behaves as if it were an infinitesimal ‘rigid rod’. Δ may also be thought of as a generic tangent vector because Microaffineness entails that it can be ‘brought into coincidence’ with the tangent to any curve at any point on it. Since we will shortly show that Δ does not reduce to a single point, it will be, so to speak, ‘large enough’ to have a slope but ‘too small’ to bend. I don't have all the answers to your questions. I am only studying the subject. Don't expect me to have all the answers if no-one else has been able to find them. I am looking for them in the books. I am not developing my own system of algebra.

Do you think the sum (x + 2dx) is using some different system of algebra than, for example, the sum (x + dx) ? I did not invent my own system of arithmetic. I am currently learning what John L. Bell has written in his book. I think you should ask the same question about what axioms of arithmetic are used in an infinitesimal approach: dx is nilsquare infinitesimal, meaning (dx)² = 0 is true, but dx=0 need not be true at the same time. So how is multiplication defined here? What axioms of arithmetic are being used? Maybe they are to be found is John L.Bell's book, he writes: "As we show in this book, within smooth infinitesimal analysis the basic calculus and differential geometry can be developed along traditional ‘infinitesimal’ lines – with full rigour – using straightforward calculations with infinitesimals in place of the limit concept. And in the 1970s startling new developments in the mathematical discipline of category theory led to the creation of smooth infinitesimal analysis, a rigorous axiomatic theory of nilsquare and nonpunctiform infinitesimals."

\[ f'(x) = \frac{f(x+dx) - f(x)}{dx}\] \[ f'(x + dx) = \frac{f(x + 2dx) - f(x + dx)}{dx}\] \[ f''(x) = \frac{df'(x)}{dx}\ = \frac{f'(x+dx) - f'(x)}{dx}\] from which, after a calculation (I skip writing this lengthy LaTeX code now, you may try it yourself), it is possible to get the result in my first post, the definition of second derivative: \[ f''(x) = \frac{f(x+2dx) - 2f(x + dx) + f(x)}{(dx)^2}\]

The question is: why can't John L. Bell's nilpotent infinitesimals possess inverses?

There is a book available, even for free download, A Primer of Infinitesimal Analysis by John L.Bell. It is possibly what I am looking for. The book says that: "A remarkable recent development in mathematics is the refounding, on a rigorous basis, of the idea of infinitesimal quantity, a notion which, before being supplanted in the nineteenth century by the limit concept, played a seminal role within the calculus and mathematical analysis."-direct quote Also an interesting note from the book: "A final remark: The theory of infinitesimals presented here should not be confused with that known as nonstandard analysis, invented by Abraham Robinson in the 1960s. The infinitesimals figuring in his formulation are ‘invertible’ (arising, in fact, as the ‘reciprocals’ of infinitely large quantities), while those with which we shall be concerned, being nilpotent, cannot possess inverses." -direct quote

I am not here to talk about those subjects. There are already enough books about them available. In the beginning, in my second post, I told that I am dealing with an infinitesimal approach: dx is nilsquare infinitesimal, meaning (dx)² = 0 is true, but dx=0 need not be true at the same time.

In my first post dx is an infinitesimal yes, I have obtained f''(x) = 2 using the definition in my first post

just to see how useful an infinitesimal approach is

Let's choose an example \[ f(x) = x^2 \] using the definition in my first post, obtain the second derivative of f(x)

it is possible to use division by zero: https://en.wikipedia.org/wiki/Signed_zero