0^0

[Albert Chan]

(11-21-2023 08:30 PM)Johnh Wrote:  On the question of what 0^0 equals, obviously it's a conundrum, but I'm sure proper mathematicians are clear on what it should be. But here's my take, and the answer is 1:

If you say that 0^0 should be the limit of x^x as x approaches zero, then you can easily see the result heading towards 1 ie:

0.1^0.1 = 0.79433
0.01^0.01 = 0.95499
0.001^0.001 = 0.99312
0.0001^0.0001 = 0.99908
0.00001^0.00001 = 0.99988 etc

And barring a typo above, then I think all calculators will agree if considered that way.

It may be OT to post to original thread, HP 48GX Collector’s Edition, so I start a new one.

You can make 0^0 be anything you want!
There are many ways to make this happen, below setup is probably simplest.

ln(x) / ln(x) = 1
ln(x) * (ln(a)/ln(x)) = ln(a)

Exponentiate both side: x ^ (ln(a)/ln(x)) = a

Note that this is an identity! No need to take limit to check.

(x → 0+) --> (ln(x) → -∞) --> (1/ln(x) → 0-)
If ln(a) is finite, we have both base and exponent approaches 0 (at different rate) --> 0^0 = a