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Math Help - Indeterminate Forms and L'Hospital's Rule

  1. #1
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    Indeterminate Forms and L'Hospital's Rule

    1. lim x->1 (x/x-1 - 1/ln x)
    This is what I got so far...
    => lim x->1 (x-1)/(x-1 ln x)
    => lim x->1 d/dx (x-1)/d/dx (x-1 ln x)
    => lim x->1 1/x ln x .....I got stuck here
    I check the book and the answer is 1/2 but how?

    2. lim x->infinity ((squareroot x^2+x)-x)
    This is what I got so far...
    => lim x->infinity d/dx (x^2+x^1/2 -x)
    => lim x->infinity d/dx (-x)/d/dx (x^2+x^-1/2) ...I got stuck here
    The answer from the book is 1/2 but how?
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  2. #2
    MHF Contributor
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    Quote Originally Posted by jsu03 View Post
    1. lim x->1 (x/x-1 - 1/ln x)
    This is what I got so far...
    => lim x->1 (x-1)/(x-1 ln x)
    I'm assuming the top line is \frac{x}{x-1}-\frac{1}{\ln(x)}

    Then you made a mistake combining these two fractions. I don't think you need to combine these fractions though, since the limit at x=1 yields \infty - \infty, which is an acceptable indeterminate form.
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