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Nov
5
comment Is $0^0=1$ postulate independent of all other axioms of complex numbers?
@JavaMan in that case his answer seems incomplete.
Nov
5
comment Is $0^0=1$ postulate independent of all other axioms of complex numbers?
@JavaMan thanks, the author claims that this leads to certain exceptions in limits laws so I asked for some examples in the comment.
Nov
4
comment Zero to the zero power - Is $0^0=1$?
0^0 is undefined by whom? I saw somebody defined it, can I now say it is defined?
Nov
4
comment Zero to the zero power - Is $0^0=1$?
"otherwise it would lead to all sorts of exceptions when dealing with the limit laws" - can you please give some examples of such exceptions?
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
So the limit of the absolute value is 1 and the limit of the argument in infinity, right?
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
there is a link in the question tinyurl.com/ckwmayc
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
Thanks, but what does mean the result which Wolfram Alpha returns?
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
@amWhy it is insane because it is neither an algebraic expression, nor indeterminacy.
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
@DonAntonio I do not want to boost the accept rate by accepting answers that do not answer my questions or by asking easy, elementary questions.
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
@did I do not know, but many of the questions I ask indeed difficult.
Nov
4
comment What is the value of $\lim_{x\to 0} x^i$?
@DonAntonio I usually ask difficult questions that little people can answer. I prefer asnwering easy questions myself.
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
@Micah this is interesting information, it would be great if you could summarize this as an answer.
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
@Micah that is I was asking about. Whether the limits exist and whether they agree in all directions.
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
well at least any smooth function of one argument can be extended to dual numbers as well following the rule: $f(a+b\varepsilon)=f(a)+f'(a)b\varepsilon$. So $p^{a+b\varepsilon}=p^a+p^a \ln p \varepsilon$ and $(a+b\varepsilon)^p=a^p+p a^{p-1} b \varepsilon$ so the both notions of exponentiation are well defined for the most of the dual numbers.
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
After all if such things would not be defined there, the set possibly would not be called a number set.
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
I see your point but I believe that the dual numbers are defined enough well so that exponentiation $x^y$ (in non-boundary cases) is defined. Am I wrong? Does square function of constant e differ in dual numbers from exp(2)?
Nov
1
comment In dual numbers, what is the value of expressions $0^\varepsilon$ and $\varepsilon^{\varepsilon}$?
@Steven Stadnicki why not? Exponentiation is defined there
Oct
30
comment A definable hyperreal system
@Ian Mateus yes this is exactly what I saw before making this question.
Oct
29
comment Is $0^0=1$ postulate independent of all other axioms of complex numbers?
You are right but this also gives it as an axiom: faculty.uml.edu/klevasseur/courses/92.421/PS1_Fall2011.pdf
Oct
29
comment Is $0^0=1$ postulate independent of all other axioms of complex numbers?
Well the definition of recursive hyper operator just postulates it to be 1 at n=3, b=0 as a special case. It is just another formulation of the same axiom.