Prove:

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

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

then $\displaystyle f(x)$ is differentiable at $\displaystyle a,$ and $\displaystyle k = f'(a).$