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 Mar 28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Arpan Banerjee logarithm is not differentiable at 0, L'Hopital's rule is not applicable Mar 28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @Arpan Banerjee counter-example: $x=1$, $\varepsilon$=2 Mar 28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @math also if $x-\varepsilon$ <0 the seond factor also becomes infinite, and I am exactly interested in the case $|x|<\varepsilon$... Mar 28 comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$ @math are u sure? What if a tends to zero from below? then the second factor in the numerator becomes infinite... Mar 28 comment Given this operator what is inverse operator? @Martin R regarding sums to non-integer limits, look here: en.wikipedia.org/wiki/Indefinite_sum anyway, I found what I was looking for. Mar 28 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. Mar 28 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? Mar 26 comment Why hyperreal numbers are built so complicatedly? Differentiability: $$(f(x+\varepsilon)-f(x-\varepsilon))/(2\varepsilon)$$ What needs more definition with it? Mar 26 comment Why hyperreal numbers are built so complicatedly? Well what questions still remain unanswered with the definition from the question? Mar 26 comment Has anybody ever considered “full derivative”? Does it mean time scales? Mar 26 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? Mar 25 comment Has anybody ever considered “full derivative”? @Kevin Carlson is this field truly hyperreal? Mar 25 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? Mar 25 comment Why hyperreal numbers are built so complicatedly? What questions about calculus remain unanswered? Mar 25 comment Has anybody ever considered “full derivative”? @Mark S. $No(\omega)$ is a hyperreal system, subfield of surreals ohio.edu/people/ehrlich/Unification.pdf Mar 25 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. Mar 25 comment General form for series coefficient of Taylor series expansion of $(x+1)^{1/x}$ @mickep around zero Mar 25 comment Has anybody ever considered “full derivative”? @Hayden in usual non-standard analysis they usually do not introduce distinguished elements $\varepsilon$ and $\omega$, thus they cannot uniquely define full derivative. Mar 25 comment Has anybody ever considered “full derivative”? @kjetil b halvorsen this is not q-derivative, totally different thing. Mar 25 comment Why hyperreal numbers are built so complicatedly? Thanks for pointing out the Levi-Civita field