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Natural Logs with Limits

  1. Oct 17, 2009 #1
    Can someone explain to me how they get last line to work out?
    I am confused on how they're using natural log with limit problems.

    Thanks for the help!

    :)
     

    Attached Files:

  2. jcsd
  3. Oct 17, 2009 #2

    lanedance

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    Homework Helper

    pics always take a while to clear, you;re better writing it up if you can
     
  4. Oct 17, 2009 #3
    Didn't realize that. Sorry. :)

    mathquestion1.jpg
     
  5. Oct 17, 2009 #4
    which equality in particular are you having a problem with? Or is it the introduction of the natural logarithm on the left-hand side?
     
  6. Oct 17, 2009 #5
    I am having trouble with why he choose to let y what it does and how we went on from there.
    How did he get

    ln ((x+3)(x+1)/(1/x))
     
  7. Oct 17, 2009 #6
    This makes each term in the sequence y(x). It is also trivial that y(x) = eln(y(x)), which, in this case, is easier to study. He made use of the logarithm rule: ln(ax) = xln(a), and algebra: x = 1/(1/x), provided x is not 0.
    Many limits are easily solved if you can find a clever rewriting of the algebraic expression like this.
     
  8. Oct 17, 2009 #7
    How did he arrive at the 2x^2 / (x+3)(x+1) step? Is there something I'm missing he did when he took the limit to infinity?
     
  9. Oct 17, 2009 #8
    The second limit gives us the indeterminate form 0/0, which makes it valid to use L'Hôpital's rule. This was the point of writing x as 1/(1/x).
     
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