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Math Help - Limit Problem

  1. #1
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    Limit Problem

    Please,
    if we have:
    lim(x to 1) ((x/(x-1)-1/lnx)
    then we get:

    lim(x to 1) ((xlnx-x+1)/(xlnx-lnx))
    then we take x=1:

    lim(x to1)((ln(1)+1-1) / (ln(1)-ln(1))
    we get 0/0
    and can apply LH:

    but NOW:
    How do we get NOW lim(x to1)((xlnx)/(xlnx+x-1)???
    What is the inversion doing here?
    Many thanks!

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  2. #2
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    Re: Limit Problem

    \lim_{x \to 1} \frac{x}{x-1} - \frac{1}{\ln{x}}

    \lim_{x \to 1} \frac{x\ln{x}}{(x-1)\ln{x}} - \frac{x-1}{(x-1)\ln{x}}

    \lim_{x \to 1} \frac{x\ln{x} - (x-1)}{(x-1)\ln{x}}

    \lim_{x \to 1} \frac{x\ln{x} - x + 1}{(x-1)\ln{x}}

    L'Hopital ...

    \lim_{x \to 1} \frac{x \cdot \frac{1}{x} + \ln{x} - 1}{(x-1)\cdot \frac{1}{x} + \ln{x}}

    \lim_{x \to 1} \frac{\ln{x}}{1 - \frac{1}{x} + \ln{x}}

    multiply numerator and denominator by x ...

    \lim_{x \to 1} \frac{x\ln{x}}{x - 1 + x\ln{x}}

    L'Hopital again ...

    \lim_{x \to 1} \frac{1+ \ln{x}}{2+\ln{x}} = \frac{1}{2}
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  3. #3
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    Re: Limit Problem

    Please...ther start all the problems...

    in the fifth line how did u get x*(1/x)+lnx-1?

    it should be d/dx xlnx-x+1, shouldn't it?

    p.s. re U using latex?
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  4. #4
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    Re: Limit Problem

    In line 5, that's the same thing as the derivative of x*lnx-x+1. If you look at the definition of L'Hopital's Rule, it implies differentiation of the numerator and the denominator. Therefore, line 5 shows the derivative of the numerator and the denominator from line 4. Line 5 simply results from using L'Hopital's Rule.
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  5. #5
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    Re: Limit Problem

    Quote Originally Posted by marijakopljar View Post
    Please...ther start all the problems...

    in the fifth line how did u get x*(1/x)+lnx-1?

    it should be d/dx xlnx-x+1, shouldn't it?

    p.s. re U using latex?
    have you not learned the product rule for derivatives ?


    \frac{d}{dx}[{\color{red}x \ln{x}} - x + 1] = {\color{red}x \cdot \frac{1}{x} + \ln{x} \cdot 1} - 1 = 1 + \ln{x} - 1 = \ln{x}


    ... and yes, the script is Latex.
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  6. #6
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    Re: Limit Problem

    Thank You, that is exactly what I missed!!!
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  7. #7
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    Re: Limit Problem

    Dear All!
    Please, could anyone help me with this:
    Find number a is element of R so that the function f(x)=a; x<=0
    ( x^1/2-1)/(x^1/3-1); x>0

    is continuous in R?

    R: a=3/2

    p.s. how to get latex here? I tried to write by tags, but it just showed the source, what I wrote, and not the nice latex text!
    Last edited by marijakopljar; December 31st 2012 at 02:03 AM.
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  8. #8
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    Re: Limit Problem

    Quote Originally Posted by marijakopljar View Post
    Dear All!
    Please, could anyone help me with this:
    Find number a is element of R so that the function f(x)=a; x<=0
    ( x^1/2-1)/(x^1/3-1); x>0

    is continuous in R?

    R: a=3/2

    p.s. how to get latex here? I tried to write by tags, but it just showed the source, what I wrote, and not the nice latex text!
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