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

  • Context: Graduate 
  • Thread starter Thread starter roger
  • Start date Start date
  • Tags Tags
    E^x Limit
Click For Summary

Discussion Overview

The discussion revolves around finding the limit of the expression (x! e^x) / (x^x * x^(1/2)) as x approaches infinity. It also touches on the nature of the factorial function and its derivative, particularly in the context of Stirling's formula and the gamma function.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Roger seeks to find the limit of (x! e^x) / (x^x * x^(1/2)) as x tends to infinity and questions whether f(x) = x! is a function and how to find its derivative.
  • Some participants reference Stirling's formula, suggesting it can be used to evaluate the limit by substituting x for n, as the function is continuous for positive x.
  • George expresses uncertainty about whether Roger's problem is intended to derive Stirling's formula.
  • Another participant notes that deriving Stirling's formula is complex and involves steps that are not immediately obvious, mentioning the presence of \sqrt{2n\pi} and complex integration.
  • Roger inquires if the limit can be found without using Stirling's formula.
  • There is a discussion about the extension of the factorial function to real numbers via the gamma function, with George explaining that \Gamma(x + 1) = x! and noting that some steps in deriving Stirling's formula are not straightforward.
  • AI confirms that the extension to real numbers means the factorial becomes continuous, allowing for the existence of a derivative, and mentions the digamma function in this context.

Areas of Agreement / Disagreement

Participants express varying levels of agreement on the use of Stirling's formula and its derivation, with some suggesting it is necessary while others question its use. The discussion about the continuity of the factorial function and its derivative also shows differing perspectives.

Contextual Notes

There are limitations regarding the assumptions made about the continuity of the factorial function and the complexity involved in deriving Stirling's formula. The discussion does not resolve these complexities or provide a definitive method for finding the limit.

roger
Messages
318
Reaction score
0
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
 
Physics news on Phys.org
Sterling's formula says limit [tex]n!=\frac{n^n}{e^n}\sqrt{2n\pi}[/tex] We need only substitute X for n, since the function is continuous for positive X, to work your problem.
 
robert Ihnot said:
Sterling's formula says limit [tex]n!=\frac{n^n}{e^n}\sqrt{2n\pi}[/tex] We need only substitute X for n, since the function is continuous for positive X, to work your problem.

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

Regards,
George
 
A standard extension of factorial to all real numbers except the negative integers is by way of the gamma function. Then [itex]\Gamma (x + 1) = x![/itex]. 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
 
thanks for your help

Can I find the limit without using sterlings formula ?
 
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 [tex]\sqrt(2n\pi)[/tex]. This frquently means the use of complex integration, but not here: http://courses.ncssm.edu/math/Stat_Inst/PDFS/appndx_1.pdf
 
Last edited:
interesting formula
 
George Jones said:
A standard extension of factorial to all real numbers except the negative integers is by way of the gamma function. Then [itex]\Gamma (x + 1) = x![/itex]. 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

Thanks for the information.

Does the extension to real numbers excluding negative, mean that the factorial becomes 'continuous' so that a derivative exists ?
 
roger said:
Does the extension to real numbers excluding negative, mean that the factorial becomes 'continuous' so that a derivative exists ?
Yes and its usually expressed in terms of digamma function.

-- AI
 

Similar threads

Undergrad A counterexample to "the integral of the limit is the limit of the integral"
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad ##f(2x)=f^2(x)-2f(x)-1/2## then find ##f(3)##
  • · Replies 17 ·
Replies
17
Views
5K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
2K
MHB What is the Limit of (3x)/(x-2) as x Approaches 2 from the Left?
  • · Replies 5 ·
Replies
5
Views
2K
MHB How to Find the Limit of (1 - x)/[(3 - x)^2] as x Approaches 3?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Is the Limit of 1/(x^2 - 9) as x Approaches -3 from the Left Positive Infinity?
  • · Replies 5 ·
Replies
5
Views
2K
High School Taking a limit and getting the wrong answer...don't know why
  • · Replies 8 ·
Replies
8
Views
2K
MHB Is the Limit of 5/(x^2 - 4) as x Approaches 2 from the Right Positive Infinity?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Explicit logical justification for last step in epsilon/delta proof?
  • · Replies 20 ·
Replies
20
Views
4K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
3K