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  1. 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, i
  2. 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}}
  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 infinitesim
  4. 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.
  5. 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.
  6. 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
  7. 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 sh
  8. \[ f'(x) = \frac{f(x+dx) - f(x)}{dx}\] \[ f'(x + dx) = \frac{f(x + 2dx) - f(x + dx)}{dx}\] \[ f''(x) = \frac{df'(x)}{dx}\ = \frac{f'(x+dx) - f'(x)}{dx}\] 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: \[ f''(x) = \frac{f(x+2dx) - 2f(x + dx) + f(x)}{(dx)^2}\]
  9. The question is: why can't John L. Bell's nilpotent infinitesimals possess inverses?
  10. 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 kn
  11. 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.
  12. In my first post dx is an infinitesimal yes, I have obtained f''(x) = 2 using the definition in my first post
  13. just to see how useful an infinitesimal approach is
  14. Let's choose an example \[ f(x) = x^2 \] using the definition in my first post, obtain the second derivative of f(x)
  15. it is possible to use division by zero: https://en.wikipedia.org/wiki/Signed_zero
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