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Math Help - Finding derivatives

  1. #1
    Senior Member I-Think's Avatar
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    Finding derivatives

    Check my work please

    Find f'(0) if
    f(x)=\frac{k(x)}{x} if x not 0
    0 if x=0

    and k(0)=k'(0)=0 and k''(0)=17

    My solution
    f'(0)=lim_{h\rightarrow{0}} \frac{f(h)-f(0)}{h}
    =lim_{h\rightarrow{0}} \frac{\frac{k(h)}{h}}{h}
    Question boils down to finding lim_{h\rightarrow{0}} k(h)

    But k is differentiable at 0, hence k is continuous at 0, so
    lim_{h\rightarrow{0}} k(h)=0
    Hence f'(0)=0
    END

    I am concerned about my solution because it doesn't use some of the information provided. Help please?
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  2. #2
    Senior Member
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    \lim \frac{k(h)}{h^2} = \lim \frac{k'(h)}{2h} = \lim \frac{k''(h)}{2} = \frac{k''(0)}{2} = \frac{17}{2}

    I applied L'Hopital's rule twice. All limits are taken as h goes to 0.
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