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X^-x

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f(x) = X to the -x power, what an unusual function. (If any enlightenment or world peace secrets come from this function, I made it)
I would love to find the limit of the function but I can't really figure it out. It is about 4.3 but, is it rational?

You should probably wait until Pi Day to post something like this.

F(x)= X^-x? I don't know much about limits, but as to if 4.3 is rational... well it certainly isn't irrational.

Since you didn't define "X", I assume "X^x" is the same as "x^x". What do you mean by "limit of the function"?

f(x) = x^-x = 1/x^x is a transcendental function. Along the positive real line, we have f(0) = 1, from where the function increases to a local maximum ~1.444667861 at x = 1/e ~= 0.3678794412. It then rapidly decays to zero as x goes to infinity. Would you like to know its structure in the complex plane too?

Pari plot:

```1.4443362 |'''x"""_''''''''''''''''''''''''''''''''''''''''''''''''''''''|
|  "     "_                                                    |
| x        x                                                   |
|_          x                                                  |
|:           "                                                 |
:             "                                                |
_              "                                               |
|               "                                              |
|                "_                                            |
|                  _                                           |
|                   _                                          |
|                    x                                         |
|                     x                                        |
|                      "_                                      |
|                        x                                     |
|                         "_                                   |
|                           x_                                 |
|                             x_                               |
|                               "x_                            |
|                                  "x_                         |
|                                     ""xx___                  |
0 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,""""xxxxxxx________
1e-07                                                          4
```
It is interesting to attempt to integrate f. The indefinite integral of x^-x is nonelementary, as can be proved with the Risch algorithm / Liouville's principle. A curious fact is that the integral from 0 to 1 of x^-x equals the sum 1 + 2^-2 + 3^-3 + 4^-4 + ... ~= 1.29128599 (search for "Sophomore's dream"). The integral from 0 to infinity is approximately 1.995455958. As far as I know, none of these constants has a closed expression in terms of familiar special functions.

You can also write x^−x as e^(−x ln(x)), −x ln(x)→−∞ when x→+∞ so its exponential→0.
Note that I don’t know where you search its limit: 0, +∞, somewhere else?

It's simple. x^-x = 1/(x^x). x^x leads to infinite, so 1/(x^x) leads to zero.

Fredrik said:

<Pretty Graph>

Do you have some sort of program that renders graphs in ASCII?

x_x

yay for liberal arts education

too.... many... variables.... ugg

Little Faith said:

Do you have some sort of program that renders graphs in ASCII?

Maple can do it, too.

damn it, i just got home from school too.

curse you, algebra 2...