# Limit of (x! e^x) / (x^x *x^1/2)

1. Jul 28, 2005

### roger

hi ,

1.)how do I find the limit of (x! e^x) / (x^x *x^1/2) as x tends to infinity ?

2.)and is f(x)= x! a function ? if so, how do I find the derivative ?

thanks for any help

Roger

2. Jul 28, 2005

### robert Ihnot

Sterling's formula says limit $$n!=\frac{n^n}{e^n}\sqrt{2n\pi}$$ We need only substitute X for n, since the function is continuous for positive X, to work your problem.

3. Jul 28, 2005

### George Jones

Staff Emeritus
I thought about saying this, but then I wondered whether the point of Roger's problem is to derive Stirling's formula.

Regards,
George

4. Jul 28, 2005

### George Jones

Staff Emeritus
A standard extension of factorial to all real numbers except the negative integers is by way of the gamma function. Then $\Gamma (x + 1) = x!$. One way to derive Stirling's formula is by using the standard integral representation of the gamma function. A couple of the steps are, however, not completely obvious.

Regards,
George

5. Jul 29, 2005

### roger

Can I find the limit without using sterlings formula ?

6. Jul 29, 2005

### robert Ihnot

When I learned about Sterling's formula it was a graduate course and the professor put the derivation on the board. It is not that simple. Note the presence of $$\sqrt(2n\pi)$$. This frquently means the use of complex integration, but not here: http://courses.ncssm.edu/math/Stat_Inst/PDFS/appndx_1.pdf

Last edited: Jul 29, 2005
7. Jul 29, 2005

### kant

interesting formula

8. Jul 30, 2005

### roger

Thanks for the information.

Does the extension to real numbers excluding negative, mean that the factorial becomes 'continuous' so that a derivative exists ?

9. Jul 30, 2005

### TenaliRaman

Yes and its usually expressed in terms of digamma function.

-- AI