- #36
Leopold Infeld
- 55
- 0
[tex]0^{0}[/tex] is what then?
I still think that it is "undefind"
I still think that it is "undefind"
arildno said:Not the program, no.
It's not a matther of how many mathematicians come up with the same thing, it's a matter of if they worked it out logically.whozum said:If a thousand mathematicians come to the conclusion that 0^0 = 1, I'm not in a position to argue with them. Perhaps we should write to Maple?
Hurkyl said:My calculator symbolically evaluates that limit to 1...
There's a more obvious counterexample:
[tex]
\lim_{x \rightarrow 0} 0^x = 0
[/tex].
The whole thing about 0^0 depends precisely what ^ means. As an operation on real numbers, 0^0 is undefined. The reason is that (0, 0) is not in the domain of ^.
Okay, so you want the "philosophical" reason. A crucial property about real operations is that they're continuous within their domain.
However, ^ cannot be continuous at (0, 0) -- the classic examples demonstrating this fact are 0^x --> 0 and x^0 --> 1 as x --> 0.
However, there are other meanings to ^. For example, there's a definition of ^ that means repeated multiplication. The empty product is, by definition, 1, so anything to the zero-th power is equal to 1. In polynomials and power series, this definition is what ^ "really" means -- that's why one uses 0^0 = 1.
Leopold Infeld said:[tex]
\lim_{x \rightarrow 0} 0^x = 0
[/tex].
Is this a possible limit?
I thought that the limit from the right isn't equal to that of the left for this case.
cuz 0^0.0000000000000001=0 cause the zillionth root of 0 is still o
but 0^-0.0000000000000001 is undefined if 0^0.0000000000000001=0 is correct.
Dont laugh guys if my arguments and terribly lame because i don't kno anything.
What has ab got to do with multiplying so many times when b is not a natrual number?mathelord said:but 0 time sany number whatever it is give 0 right
There isn't any number log(0)mathelord said:but 0 time sany number whatever it is give 0 right
Although perhaps their proof only covered that it's also true that [itex]\lim_{x \rightarrow 0^-} x^x = 1[/itex]Hurkyl said:That's (almost) correct; [itex]\lim_{x \rightarrow 0^+} x^x = 1[/itex].
Zurtex said:Although perhaps their proof only covered that it's also true that [itex]\lim_{x \rightarrow 0^-} x^x = 1[/itex]
[tex]\left( -\frac{1}{2} \right)^{-\frac{1}{2}} = -i \sqrt{2}[/tex]HallsofIvy said:What is true is that xx is not defined for negative x so that doesn't even make sense.
arildno said:Note that we may approach [tex]0^{0}[/tex] in the following manner, by letting x go to 0 from the positive side:
[tex]f(x)=(e^{-\frac{1}{x}})^{\alpha{x}}[/tex]
But, evidently, we have [tex]f(x)=e^{-\alpha}[/tex]
In this case therefore, we have [tex]\lim_{x\to0}f=e^{-\alpha}[/tex]