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April 9th, 2018, 06:36 PM   #1
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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.
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April 9th, 2018, 06:40 PM   #2
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Any chance you could typeset your equations? It's hard to read.
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April 9th, 2018, 06:56 PM   #3
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Quote:
Originally Posted by Joppy View Post
Any chance you could typeset your equations? It's hard to read.
I tried to make it as nice as I could.
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April 9th, 2018, 07:43 PM   #4
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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.
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April 9th, 2018, 07:45 PM   #5
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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
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April 9th, 2018, 07:47 PM   #6
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Quote:
Originally Posted by SDK View Post
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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