Unusual Limit involving e

  • #1
34
0
This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved [tex]\int_{0}^{\infty}x e^{-x}dx=1.[/tex]

Well, when we do Integration by parts we will have [tex]let[/tex] [tex] u = x [/tex] [tex] du = dx [/tex] [tex]dv = e^{-x}dx[/tex] [tex]v = -e^{-x}[/tex] so that would give us [tex]\int_{0}^{\infty}x e^{-x}dx= - x e^{-x} + \int_{0}^{\infty} e^{-x}dx [/tex] evaluating the second term will just yield 1, so, [tex]\int_{0}^{\infty}x e^{-x}dx= - x e^{-x} + 1,[/tex] the second term should yield zero by employing limits as x approaches infinity, [tex] \lim_{x\rightarrow \infty }xe^{-x}=0 [/tex] but whoa, it didn't. I will yield instead [tex]1/e[/tex] Where did I went wrong?
 

Answers and Replies

  • #2
34,017
5,672
This was just very basic, I have accepted it in just a heartbeat, but when I tried to chopped it and examined one by one, somethings fishy is happening, this just involved [tex]\int_{0}^{\infty}x e^{-x}dx=1.[/tex]

Well, when we do Integration by parts we will have let [tex] u = x [/tex] [tex] du = dx [/tex] [tex]dv = e^{-x}dx[/tex] [tex]v = -e^{-x}[/tex] so that would give us [tex]\int_{0}^{\infty}x e^{-x}dx= - x e^{-x} + \int_{0}^{\infty} e^{-x}dx [/tex] evaluating the second term will just yield 1, so, [tex]\int_{0}^{\infty}x e^{-x}dx= - x e^{-x} + 1,[/tex] the second term should yield zero by employing limits as x approaches infinity, [tex] \lim_{x\rightarrow \infty }xe^{-x}=0 [/tex] but whoa, it didn't. I will yield instead [tex]1/e[/tex] Where did I went wrong?
It's not clear what you're asking.
##\lim_{x \to \infty} xe^{-x} = 0## is correct. Are you getting 1/e for this limit? If so, that's where you went wrong.
 
  • #3
34
0
I get it now, I am imputing the wrong variable from my computer, indeed lim xe^{-x} as x approaches infinity indeed gives 0.
 

Related Threads on Unusual Limit involving e

Replies
1
Views
2K
Replies
5
Views
5K
Replies
2
Views
945
Replies
3
Views
731
Replies
1
Views
4K
Replies
1
Views
1K
  • Last Post
Replies
7
Views
3K
  • Last Post
Replies
9
Views
3K
Replies
4
Views
7K
  • Last Post
Replies
2
Views
650
Top