# Math Help - Best approximation

1. ## 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 $k = f'(a).$

2. Originally Posted by cgiulz
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.