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 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(n-n)) = 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
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 Originally Posted by Joppy Any chance you could typeset your equations? It's hard to read.
I tried to make it as nice as I could.

 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 $0 = n-n$ then it is still true that $\lim_{n \to \infty} n-n = 0$ 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. Thanks from SDK
April 9th, 2018, 07:47 PM   #6
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 Originally Posted by SDK 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 $0 = n-n$ then it is still true that $\lim_{n \to \infty} n-n = 0$ 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.
So you are saying that lim [(x+k)-(x)] = lim (k) = k while lim (x+k) - lim (x) dne. Very interesting. This point never came up before for me. Thanks for your time.

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