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Math Help - help differentiable function

  1. #1
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    help differentiable function

    if f is differentiable in f(x_{0}) so: lim_{h\rightarrow0}\frac{f(x_{0}+h)-f(x_{0}-h)}{2h}=f'(x_{0})
    i need to know if this is right or false... i tried to use arithmetic rules on this but i got nothing..
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  2. #2
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    Re: help differentiable function

    It is corect! since lim [f(x0+h)-f(x0-h)]/h = 2f'(x0)

    to prove this get the limit definition of the derivative....lim[f(x0+h)-f(x0)]/h = f'(x0) and lim[f(x0-h)-f(x0)]/h = -f'(x0)
    it is easy..
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  3. #3
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    Re: help differentiable function

    If f is differentiable, then the given limit exists and is what you say it is, so you are correct. The converse is not true, though. The limit can exist for functions that are not differentiable - take f(x)=\frac{1}{x^2} and x_0=0, for example.

    - Hollywood
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