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Math Help - Help Solving a limit

  1. #1
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    Help Solving a limit

    Hey everyone,

    Can anyone help solve this limit? I'm having some difficulty.



    Thanks
    Last edited by evant8950; January 22nd 2010 at 07:45 AM.
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  2. #2
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    Hello evant8950

    Welcome to Math Help Forum!
    Quote Originally Posted by evant8950 View Post
    Hey everyone,

    Can anyone help solve this limit? I'm having some difficulty.



    Thanks
    I assume you mean:
    \lim_{x\to0}\left(\frac{1}{x\sqrt{1+x}}-\frac1x\right)
    In which case write it as:
    \frac{1}{x\sqrt{1+x}}-\frac1x =\frac1x\Big((1+x)^{-\frac12}-1\Big)
    Now write the Binomial expansion of (1+x)^{-\frac12}, and simplify. Then you'll find that
    \lim_{x\to0}\left(\frac{1}{x\sqrt{1+x}}-\frac1x\right)=-\frac12
    Grandad
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  3. #3
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    Quote Originally Posted by evant8950 View Post
    Hey everyone,

    Can anyone help solve this limit? I'm having some difficulty.



    Thanks
    For  <br />
\lim<br />
_{t \to 0}<br />
\frac{1 - \sqrt{1+t}}{t \sqrt{1+t}} <br />
and multiplying by  <br />
\frac{1 + \sqrt{1+t}}{1 + \sqrt{1+t}}<br />
gives

     <br />
\lim_{t \to 0} \frac{-1}{(1+\sqrt{1+t})\sqrt{1+t}}<br />

    I think you can finish.
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  4. #4
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    Quote Originally Posted by evant8950 View Post
    Hey everyone,

    Can anyone help solve this limit? I'm having some difficulty.



    Thanks

    \frac{1}{t\sqrt{1+t}}-\frac{1}{t}=\frac{1-\sqrt{1+t}}{t\sqrt{1+t}}= \frac{-t}{t\sqrt{1+t}\left(1+\sqrt{1+t}\right)} =-\frac{1}{\sqrt{1+t}\left(1+\sqrt{1+t}\right)}\xrig  htarrow[t\to 0]{} ...?

    Tonio


    Oh, it never mind: somebody already did ALL the work for you. **sigh**
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  5. #5
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    Mmm..

    We have \frac{1}{t\sqrt{1+t}}-\frac{1}{t}=\frac{1}{t}\left( \frac{1-\sqrt{1+t}}{\sqrt{1+t}} \right)=-\frac{1}{\left( 1+\sqrt{1+t} \right)\sqrt{1+t}}, for t\ne0, now as t\to0 yields the result.

    (Ahhh, I'm slow today.)
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  6. #6
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    Thanks everyone for the help. I appreciate it.
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  7. #7
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    x=Root(t+1)

    so =(1-x)/(-x)(x+1)(1-x)=1/(-x^2-x)

    t->0 means x->1

    lim 1/(-x^2-x)=-0.5
    x->1

    OK..
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  8. #8
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    oops..I came late..
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