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L'Hopital's rule

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Find lim x->1+ of [(1/(x-1))-(1/lnx)].

Here is my work...
=(lnx-(x-1)) / ((x-1)(lnx))

=(lnx-1) / (lnx+ (x+1)/x)

This becomes(1/x) / ((1/x)+(1/x^2))

which becomes 1/ (1/x^2)
This equals 1/2. I understand the answer has to be -1/2, but I am having trouble figuring out where that negative sign is coming from. What have I missed? Thank you in advance for your assistance.

  • L'Hopital's rule -

    =(lnx-(x-1)) / ((x-1)(lnx))

    How did you get to the next line?

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