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What's the difference between [tex]f(x)=3[/tex] and [tex]f(x)=3x^0[/tex] ? and why Limit of the second function when [tex]x\rightarrow0[/tex] exists ? and is the second function continuous at [tex]x=0[/tex] ?
I disagree very much with the mathematician's explanation. They're like saying that ##0^0 = 1## is a consensus among mathematicians, it's not.Here's a fun discussion on the 0^0 from the perspective of students, teachers, and mathematicians:
http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to-the-zeroth-power-equal-why-do-mathematicians-and-high-school-teachers-disagree/
Maybe it's not. But I also think it's natural, simply for the reason that taking something to a power is something multiplicative. And therefore it makes sense to define it as the unit 1.I disagree very much with the mathematician's explanation. They're like saying that ##0^0 = 1## is a consensus among mathematicians, it's not.
Sure, there are plenty of reasons why it should be ##1##. The best are set theoretic and category theoretic. But there is no consensus.Maybe it's not. But I also think it's natural, simply for the reason that taking something to a power is something multiplicative. And therefore it makes sense to define it as the unit 1.
At least its a very binary decision and mathematicians can wear the shirt:Sure, there are plenty of reasons why it should be ##1##. The best are set theoretic and category theoretic. But there is no consensus.
Binary? What about defining 0^{0} as the limit $$\lim_{x \to 0} x^{1/\log(x)} = e$$? You can also get every other number of course.At least its a very binary decision and mathematicians can wear the shirt:
It all reduces to 1's and 0's somehow. :-)Binary? What about defining 0^{0} as the limit $$\lim_{x \to 0} x^{1/\log(x)} = e$$? You can also get every other number of course.
The thing is though that ##0^0 = 1## in this case only when the exponent is already seen as an integer. If the exponent is seen as a real number then it's better to leave ##0^0## undefined. So we have this weird context-dependent rule: ##y^x = 1## if ##x## is an integer and undefined when it can take on a continuous range of variables. Such a definition would be the most interesting one, but have no idea how to formalize that in a neat way. Maybe some kind of typed logic or something.The vast, boring, senseless, stupid, Sisyphus-like, unprofitable and endless job to rewrite formerly beautiful and nice formulas like, e.g.
$$\exp(x) = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$
is justification enough for the choice of ##1##.