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Math Help - Solving

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    71

    Solving

    Define a function f by f(x)=x+2ln(x) and let g(x)=f(x)-A-B(x-1)-C(x-1)^2. Find the values of A, B, and C so that L=lim x→1 g(x)/(x-1)^3 exists and is finite.
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  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    408

    L'Hospital

    L'Hospital's Rule states that a limit L=\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)} if the limit is indeterminate, that is, \frac{f(a)}{g(a)}=\frac00

    Here, if we substitute x=1 into the function \frac{x-\ln x-A-B(x-1)-C(x-1)^2}{(x-1)^3}, we get \frac{1-A}{0}. Notice that if the numerator is NOT zero, then the limit will run off to \pm\infty. Therefore, it is necessary that A=1 so the limit is indeterminate, otherwise the limit will not converge at all. Apply L'Hospital's Rule...

    Now L=\lim_{x\to1}\frac{1+2/x-B-2C(x-1)}{3(x-1)^2}, so repeat the process to find B. Keep going and you can get A,B,C, and also the final value of the limit L.
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