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Homework Help: Limit is e

  1. Oct 31, 2005 #1
    ok I have this limit question that was done in class but i didn't catch it at the time but they grazed over a step where i'm not sure what the reasoning was.
    [tex] \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n} = \lim_{n \rightarrow \infty} \left( 1 + \frac{ \frac{1}{n} }{ 1 + \frac{1}{n} } \right)^{n} = e [/tex]
    I see the multiplication of one in the form of 1/n over 1/n and i know that [tex] \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n} = e [/tex] and i can see the similarity and possible substituions....but what happens to the 'n' exponet since it doesn't substitute nicely?
     
    Last edited: Oct 31, 2005
  2. jcsd
  3. Oct 31, 2005 #2

    Galileo

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    I see no reason to divide the numerator and the denominator by n. You can simply use a substitution like m=n+1. Or write:
    [tex]\lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n} = \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n + 1} \right)^{n+1}\cdot \frac{1}{1+\frac{1}{n+1}}[/tex]
     
  4. Nov 1, 2005 #3
    I think I'm a little more confused now....I do agree that i dont see what dividing by n did to help.
     
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