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Math Help - defining differentiability

  1. #1
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    defining differentiability

    I have this question on a mock exam paper for my university course; does anyone have any ideas of what to do?

    The definition of differentiability is as follows. A function f:R --> R that is continuous in some interval around a point x=a is differentiable at x=a and has derivative f'(a) if the following limit exists,

    f'(a)=lim((f(x)-f(a))/(x-a))

    use this definition to prove that if two functions g:R-->R and h:R-->R are differentiable at the point x=a so is the function Ag+Bh where A and B are constants
    Last edited by CaptainBlack; August 5th 2009 at 11:47 AM.
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  2. #2
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     \lim_{x \to a} \frac {(Ag+Bh)(x) - (Ag+Bh)(a)}{x-a} = \lim_{x \to a} \frac{Ag(x) + Bh(x) - Ag(a) - Bh(a)}{x-a}


     = \lim_{x \to a} \frac{Ag(x)-Ag(a)}{x-a} + \lim_{x \to a} \frac{Bh(x)-Bh(a)}{x-a}


     = A \lim_{x \to a} \frac{g(x)-g(a)}{x-a} + B \lim_{x \to a} \frac{h(x)-h(a)}{x-a} = Ag'(a) + Bh'(a)
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  3. #3
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    Thank you
    Last edited by Lupus; August 5th 2009 at 07:39 AM.
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  4. #4
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    i take it that this is a harder question then i thought.
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