defining differentiability

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August 2nd 2009, 06:26 AM
Lupus

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
August 2nd 2009, 07:00 AM
Random Variable

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

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

$= A \lim_{x \to a} \frac{g(x)-g(a)}{x-a} + B \lim_{x \to a} \frac{h(x)-h(a)}{x-a} = Ag'(a) + Bh'(a)$
August 3rd 2009, 08:49 AM
Lupus

Thank you
August 5th 2009, 05:46 AM
Lupus

i take it that this is a harder question then i thought.