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Mar
28
comment Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
@Martin R as $\log |a|$
Mar
28
revised Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
deleted 1 character in body
Mar
28
revised Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
added 180 characters in body
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@Timbuc en.wikipedia.org/wiki/Non-standard_analysis
Mar
28
comment Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
@abel Cosine integral mathworld.wolfram.com/CosineIntegral.html
Mar
28
asked Proof that $\lim_{x\to 0^+}{\sin \frac1x}=\sin \left(\frac{1}{2}\right)-\frac 12 \text{Ci}\left(\frac{1}{2}\right)$
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 we get infinity in both numerator and denomenator with this rule. If u know how to apply it properly, make an answer please.
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
asked Find limit $\lim_{a\to 0} \, \frac{\left(a^{2 \varepsilon }-1\right) a^{x-\varepsilon }}{2 \varepsilon \log (a)}$
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
answered Given this operator what is inverse operator?
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
28
asked Given this operator what is inverse operator?
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?