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Can't find limit

  1. Sep 15, 2009 #1

    honestrosewater

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    Gold Member

    Problem
    Find the limit.

    [tex]
    \lim_{x \rightarrow 0} \frac{\frac{1}{x + 1} - 1}{x}
    [/tex]

    My attempt
    I'm totally stuck. domain(f) = R - {0}. Setting g: R - {-1} --> R, g(x) = 1/(x+1) - 1 and h: R --> R, h(x) = x, g and h are both continuous, but g(0) = 0 and h(0) = 0. It looks like f(0) is just a hole. Perhaps I will try factoring again. My book says the limit is -1, but I don't see how it expects me to find it.
     
  2. jcsd
  3. Sep 15, 2009 #2

    tiny-tim

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    Science Advisor
    Homework Helper

    Hi honestrosewater! :smile:

    erm :redface:

    what's 1/(x+1) - 1 ? :wink:

    ( alternatively, use l'Hôpital's rule )
     
  4. Sep 15, 2009 #3

    honestrosewater

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    Gold Member

    1/(x + 1) - 1 = -x/(x+1)? Does that help me somehow? Sorry, I am sure it's something simple, but I cannot see it. Oh... right.

    [tex]\left(\frac{-x}{x + 1}\right)\left(\frac{1}{x}\right) = \frac{-1}{x + 1} = g(x)
    [/tex]

    g is continuous and x != 0 implies g(x) = f(x), so lim(x --> 0) f(0) = lim(x --> 0) g(0) = -1. Hah.

    Thanks for this tip. I will remember it later. Unfortunately, passing my test means applying the algorithms that the book has taught us, and we have not covered derivatives yet or been taught that rule. I have already been warned about using theorems that I am not supposed to know.

    Thanks! :^)
     
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