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Math Help - Limit at infinity involving e

  1. #1
    Member nautica17's Avatar
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    Limit at infinity involving e

    I'm having some troubles with this one:

    lim x -> +infinity of (x + 1)^x / (x^x)

    The answer in the back of the book says the limit is e. So I figure that somehow I need to get the denominator to be 1 and the numerator to be in the form (1 + 1/x)^x. But I can't seem to get that. Is there some other way? Or maybe I'm missing a step. I've tried dividing top and bottom by x and x^x but this x exponent is messing me up algebraically.

    Any help, please?
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  2. #2
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    Quote Originally Posted by nautica17 View Post
    I'm having some troubles with this one:

    lim x -> +infinity of (x + 1)^x / (x^x)

    The answer in the back of the book says the limit is e. So I figure that somehow I need to get the denominator to be 1 and the numerator to be in the form (1 + 1/x)^x. But I can't seem to get that. Is there some other way? Or maybe I'm missing a step. I've tried dividing top and bottom by x and x^x but this x exponent is messing me up algebraically.

    Any help, please?
    \frac{(x+1)^x}{x^x} = \left( \frac{x+1}{x} \right) ^x = \left( 1+ \frac{1}{x} \right) ^x
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  3. #3
    Member nautica17's Avatar
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    Quote Originally Posted by pomp View Post
    \frac{(x+1)^x}{x^x} = \left( \frac{x+1}{x} \right) ^x = \left( 1+ \frac{1}{x} \right) ^x
    How did I not see that? Okay, thank-you that makes sense.
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  4. #4
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    Quote Originally Posted by nautica17 View Post
    How did I not see that? Okay, thank-you that makes sense.
    No worries. You were probably just overcomplicating things, happens to all of us and it happens far too often believe me.
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