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Homework Help: If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h

  1. Jul 21, 2010 #1
    1. The problem statement, all variables and given/known data

    If f is differentiable at x = a, evaluate lim[h->0] (f(a+2h)-f(a+3h))/h


    2. Relevant equations

    We know that f'(a) = lim[h->0] (f(a+h)-f(a))/h

    3. The attempt at a solution

    I have done the following, and I am not sure if it is correct, though the result makes sense intuitively:

    lim[h->0] (f(a+2h)-f(a+3h))/h

    = 2* lim[h->0] (f(a+2h)-f(a+3h))/ (2*h)

    = 2* lim[h->0] (f(a+2h)-f(a)-f(a+3h)+f(a))/ (2*h)

    = 2* { lim[h->0] (f(a+2h)-f(a))/(2*h) } - 2*{lim[h->0] f(a+3h)-f(a))/ (2*h)

    And here is part about which I am unsure, since I am working with multiples of h:

    = 2*f'(a) - 3*{lim[h->0] f(a+3h)-f(a))/ 3*h)

    = 2*f'(a) - 3*f'(a) = -f'(a)
     
  2. jcsd
  3. Jul 21, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    Looks fine to me.
     
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