April 9th, 2018, 06:36 PM  #1 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  Limit question
It can be shown that Lim(n>oo) [(1+1/n)^n*n en] = e/2 (and I can do this) Now consider this method: (note all limits are as x goes to infinity) Lim(n>oo) [(1+1/n)^n*n  en] = Lim(1+1/n)^n * Lim(n) Lim(e)* Lim(n) = e*Lim(n)  e*Lim(n) = e(Lim(n)  Lim(n)) =e(Lim(nn)) = eLim(0) = e*0 = 0 Where is the mistake?? Last edited by Jomo; April 9th, 2018 at 07:01 PM. 
April 9th, 2018, 06:40 PM  #2 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,834 Thanks: 650 Math Focus: Yet to find out. 
Any chance you could typeset your equations? It's hard to read.

April 9th, 2018, 06:56 PM  #3 
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  
April 9th, 2018, 07:43 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 635 Thanks: 401 Math Focus: Dynamical systems, analytic function theory, numerics 
It goes off the rails in line 1. The following is not true: \[\lim_{n \to \infty} (1+1/n)^n n  en =\lim_{n \to \infty} (1+1/n)^n \cdot \lim_{n \to \infty} n  e \lim_{n \to \infty} n \] The problem is that two of the limits on the right don't exist (specifically, they are unbounded). The fact is, you can't just arbitrarily break limits apart and you must be more careful than that. For a simple example, consider $\lim_{n \to \infty} 0 = 0$ which should be obvious. However, if I write $0 = nn$ then it is still true that $\lim_{n \to \infty} nn = 0$ but it isn't true that this limit is $\lim_{n \to \infty} n  \lim_{n \to \infty} n$ since neither of these limits exists. 
April 9th, 2018, 07:45 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2654 Math Focus: Mainly analysis and algebra 
We appear to be talking about $$\lim_{n \to \infty} \left( n \left(1+\tfrac1n\right)^n en\right)$$ But your first step doesn't make sense. Two of the limits in the expression don't exist. 
April 9th, 2018, 07:47 PM  #6  
Newbie Joined: Nov 2013 Posts: 28 Thanks: 8  Quote:
 

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