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 Mar28 asked Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ Mar28 comment Given this operator what is inverse operator? @Martin R 4 well the limit is added for some functions that have t in the denomenator (like 1/x). The sum should be computerd in closed form then take limit. The limit is not necessary for other functions. Mar28 answered Given this operator what is inverse operator? Mar28 comment Given this operator what is inverse operator? @Olivier Oloa actually, $$\Delta_{sym}[f(x)]=(\Delta_{full}[f(x)]+\Delta_{full}[f(x-\varepsilon)])/2$$, but what does it help? Mar28 asked Given this operator what is inverse operator? Mar26 comment Why hyperreal numbers are built so complicatedly? Differentiability: $$(f(x+\varepsilon)-f(x-\varepsilon))/(2\varepsilon)$$ What needs more definition with it? Mar26 comment Why hyperreal numbers are built so complicatedly? Well what questions still remain unanswered with the definition from the question? Mar26 comment Has anybody ever considered “full derivative”? Does it mean time scales? Mar26 comment Has anybody ever considered “full derivative”? @Mark S. what a problem in defining $\sin \varepsilon$? It just can be represented as a series or in closed form... Where the problem is? Mar25 revised Has anybody ever considered “full derivative”? edited title Mar25 comment Has anybody ever considered “full derivative”? @Kevin Carlson is this field truly hyperreal? Mar25 revised Has anybody ever considered “full derivative”? added 565 characters in body Mar25 comment Has anybody ever considered “full derivative”? @Mark S. so basically if to add to rationals $\varepsilon$ as in this post we get Levi-Civita field? Mar25 comment Why hyperreal numbers are built so complicatedly? What questions about calculus remain unanswered? Mar25 comment Has anybody ever considered “full derivative”? @Mark S. $No(\omega)$ is a hyperreal system, subfield of surreals ohio.edu/people/ehrlich/Unification.pdf Mar25 revised Has anybody ever considered “full derivative”? added 22 characters in body Mar25 revised Has anybody ever considered “full derivative”? added 276 characters in body Mar25 awarded Nice Question Mar25 comment Has anybody ever considered “full derivative”? @Hayden I am about introducing one element, $\varepsilon$, similarly to how complex $i$ introduced. One can argue such system would be undefinable, but then complex numbers are also undefinable because $i$ is indistinguishable from $-i$. By the way, in the context of surreals, hyperreal numbers can be considered a subfield $No(\omega)$. Then this $\omega$ has definite meaning: it is considered equal to the first infinite ordinal. Mar25 revised General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$ added 277 characters in body