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Math Help - Best approximation

  1. #1
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    Best approximation

    Prove:

    If f(x) is defined for x \approx a, and there is a number k such that

    f(x) = f(a) + k(x - a) + e(x), where \displaystyle\lim_{x \to a}\frac{e(x)}{x - a} = 0,

    then f(x) is differentiable at a, and k = f'(a).
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  2. #2
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    Quote Originally Posted by cgiulz View Post
    Prove:

    If f(x) is defined for x \approx a, and there is a number k such that

    f(x) = f(a) + k(x - a) + e(x), where \displaystyle\lim_{x \to a}\frac{e(x)}{x - a} = 0,

    then f(x) is differentiable at a, and k = f'(a).
    My first thought was "what is your definition of 'differentiable'?". Can we take it that this is in R and we are using the Calculus I definition of the derivative: f'(a)= \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}? If so, just put x= a+h in your formula for f(x) and put that into the difference quotient.
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