Question

Question #3168f

Answer 1

I don't believe you should've applyied L'H's rule since you would've been left with `oo^oo`.

Answer 2

No you cannot apply l'Hopital infinitely many times.

Explanation:

One problem with your approach is that the derivative of `x^x` cannot be found by the power rule. (It is not a power function.) We differentiate `x^x` by some version of logarithmic differentiation.

`x^x = e^(xlnx)`

so `d/dx (x^x ) = x^x(1+lnx)`

and the second derivative is

`d^2/dx^2 (x^x ) = x^x(1+lnx)^2+(x^x)/x`.