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Math Help - limit involving exponential

  1. #1
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    limit involving exponential

    Hello,

    I've lurked this forum before but this is my first time posting. I came across a limit this morning that I feel like I really SHOULD be able to answer but can't. I think I'm missing something stupid. Especially since I think this particular limit comes up a bunch.

    <br /> <br />
\lim_{x \to \infty} (1 + \frac{1}{x})^x<br /> <br />

    I've found out that it equals e (from online calculators ) but I have no idea why. I've tried converting it into a rational and using l'Hopital's rule, but I'm getting stuck. Am I on the right track? Any hints?
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  2. #2
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    Quote Originally Posted by erikzb View Post
    I've tried converting it into a rational and using l'Hopital's rule, but I'm getting stuck.
    Please, post your work.
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  3. #3
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    If we rewrite the limit, this may help.

    [(x+1)/x]^x

    Now lets take the ln and raise to the e

    e^{xln[(x+1)/x]}

    e^x*[(x+1)/x]

    The limit of [(x+1)/x]= x/x=1

    Now we have e^x=e^1=e
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  4. #4
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    Thanks. I definitely was doing something stupid. I changed the function like in your first step, but then took both the top and bottom the the power x, and changed THOSE to exponentials, and tried using L'H. I'm a crazy person.
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