1. If 0^0 is defined it could logically be defined as the limit of x^0 as x
approaches 0, which is 1.
2. If 0^0 is defined it could logically be defined as the limit of 0^x as x
approaches 0 which is 0.
To most precalc students (intuitively looking at limits) this seems to be
enough to convince them that 0^0 is undefined.
3. If 0^0 is undefined then it could logically be defined as limit of x^x
as x approaches 0.
At this point I usualy graph y = x^x and get some interesting discussions as
to why there are some points with negative values of x which are defined,
but at most values y is undefined (or at least not graphed). They also
notice that the right hand limit appears to be 1. It's a real good way to
use logarithmic differentiation to prove this.
Then I usually ask my students, "Which is larger e^pi or pi^e? And what does
that have to do with this graph?" We guess, and I don't let them use the
calculators to estimate the answer until after everyone guesses. then, by
finding the minimum (with log diff) occurs at 1/e we use a little algebra to
answer the question.
Anyway, I think this is a great function to incorporate the calculator to
explore a simple concept (definition of 0^0)and then go beyond what the
calculator is able to do and discuss limits, complex variables, one sided
limits, logarithmic differentiation, and finally to see that some "simple"
questions aren't so simple after all.
At 05:57 PM 7/16/98 PDT, you wrote:
>>
>>but 0^0 is still evaluated to be one!!!
>>
>
>
>As well it should be! Although some analysts seem timid about defining
>0^0, logicians and set theorists are not. Defining 0^0 as 1 is not some
>pecularity, but is actually fairly natural. I believe Paul Halmos's
>book, Naive Set Theory, was the first place I saw it. So if my
>explanation below is opaque, you can consult his book.
>
>Exponentiation is defined first on the non-negative integers: n^m is
>the size of the set of all functions from a set of size m to a set of
>size n . So for example, there are 2^3 functions from the set {0,1,2}
>to the set {0,1}.
>
>Now n^0 should be the number of functions (i.e., sets of ordered
>pairs...) from a set of size 0 to a set of size n. There is exactly one
>set that has an ordered pair for each element in the (empty) domain,
>namely the empty set. Thus n^0 is 1 for any non-negative integer n. In
>particular, 0^0 = 1.
>
>I think the reluctance to accept this definition stems largely from the
>fact that the function f(x,y) = x^y has no limit at the origin, so in
>particular is not continuous at the origin. Thus when evaluating limits
>with the indeterminate form 0^0, the result might well be different from
>1. If we taught our students that 0^0 = 1, we would probably have even
>more difficulty than now in preventing them from "plugging in" to
>evaluate limits of g(x)^(h(x)) when both g and h go to zero.
>
>We've probably all seen numerous articles and/or texts that define power
>series of a real or complex variable with sigma notation, the sum from n
>= 0 to infinity of a_n * z^n . When z = 0, the value of the power
>series is a_0. We are implicitly accepting that z^0 = 1 for all z, not
>just all non-zero values of z. This is a notational convention rather
>than a definition, but the definition exists and is useful even outside
>of the realm of logicians.
>
>Personally, I'm quite satisfied that the TI-92's evaluates 0^0 as 1.
>
>Bruce Yoshiwara
>Los Angeles Pierce College
>
>>>
>>Lawrence G. Gilligan
>>Professor of Mathematics
>>
>>University of Cincinnati
>>OMI College of Applied Science
>>2220 Victory Parkway
>>Cincinnati, OH 45206
>>
>>
>>(513) 556-4868
>>
>>FAX: (513) 556-4878
>>
>>http://www.uc.edu/~gilligan
>>
>
>
>
>
>
>----------------------------------------------------------------------
>byoshiwara@hotmail.com
>http://www.lapc.cc.ca.us/usr/yoshibw
>
>"...and I have assuredly found an admirable proof of this, but the
>margin is too narrow to contain it." --Pierre de Fermat
>----------------------------------------------------------------------
>
>
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Bret Taylor Lake-Sumter Community College Leesburg FL
"It matters not the subject taught, nor all the books on all the shelves.
What matters more, yes most of all, is what the teachers are themselves."
John Wooden
John 3: 3^3 + 3
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