..It's been a while since I've taken Calc II, so I'm having trouble remember the exact steps to solve a problem like this:
limit as x goes to infinity of x!/(x^x).
I know that the bottom grows faster than the top Mr F says: Therefore it should be clear that the limit is equal to zero.
and that the limit is nonzero, Mr F says: NO it's not. See above.
but I don't remember how to get to that answer: Mr F says: Perhaps you should consider the gamma function ..... Furthemore, when x is an integer the analysis is not too difficult.
There are n factors on the top: n * (n-1) * (n-2) * ....* (n-(n-1)) * 1
And there are n on the bottom: n * n * n .... * n
Every factor on the top is >= n, so if you rewrite the division you get:
(n/n) * ((n-1)/n) * ((n-2)/n) ... * (1/n)
Half of the terms are less than 1/2 and half of the terms are greater than 1/2. Intuitively I want to say that as N goes to infinity, it's a wash and the limit is 1/2, but I don't know how to show that mathematically.