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Limit similar to differentiation

  1. Oct 29, 2008 #1
    1. The problem statement, all variables and given/known data

    f a differentiable real valued function

    lim h - > 0 of (f(x + ah) - f(x + bh))/h

    where a,b real numbers

    2. Relevant equations

    definition of derivative

    lim h-> 0 of (f(x+h) - f(x))/h

    3. The attempt at a solution

    I've picked several functions like x^2 and 1/2x

    in the first case you get 2xa - 2xb = 2x(a-b)

    in the latter you get 1/2a - 1/2b = 1/2(a-b)

    This leads me to suspect f'(x)*(a - b) as the solution, I just have no idea what to do to the limit.

    I tried working backwards from my solution, but I still am missing something. Any help is appreciated.
     
  2. jcsd
  3. Oct 29, 2008 #2

    statdad

    User Avatar
    Homework Helper

    See if this gets you anywhere.

    [tex]
    \frac{f(x+ah)-f(x+bh)}{h} = \frac{f(x+ah)-f(x)}{h} + \frac{f(x) - f(x+bh)}{h} = a\left(\frac{f(x+ah)-f(x)}{ah}\right) - b \left(\frac{f(x) - f(x+bh)}{bh}\right)
    [/tex]
     
  4. Oct 29, 2008 #3
    thanks, that last one since you are multiplying by -b the numerator should be f(x +bh) - f(x)

    It is clear that as h -> 0 ah and bh ->0 and therefore those we are left with f'(x)

    Thank you so much for your help.
     
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