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Math Help - L'hopital's rule

  1. #1
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    Exclamation L'hopital's rule

    lim [((x)/(x-1))-((1)/(ln x))]
    x->1+

    use l'hopitals rule
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  2. #2
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    Subtract the fractions to get an indeterminate form: \lim_{x \to 1^+} \left( \frac{x}{x-1} -  \frac{1}{\ln x}\right)  = \lim_{x \to 1^+} \frac{{\color{red}x\ln x} - {\color{blue}x} + 1}{{\color{magenta}(x-1)\ln x}} = \left[\frac{0}{0}\right]

    Use L'hopital's rule to get (product rule is needed to take care of the xlnx): = \lim_{x \to 1^+} \frac{ {\color{red}\ln x + 1} - {\color{blue}1}}{{\color{magenta}\ln x + 1 -\frac{1}{x}}} = \lim_{x \to 1^+} \frac{\ln x}{\ln x + 1 -\frac{1}{x}} = \left[\frac{0}{0}\right]

    This is another indeterminate form so apply L'hopital's rule again. Try taking the limit and if you get a definite answer then you're done.
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