$\displaystyle (a)$ Prove if $\displaystyle f'(a)$ exists, then $\displaystyle f(x) = L(x) + e(x),$ where $\displaystyle L(x) = f(a) + f'(a)(x - a)$ and $\displaystyle \displaystyle\lim_{x\to a}\frac{e(x)}{x - a} = 0.$

$\displaystyle (b) L(x)$ is called the linearization of $\displaystyle f(x)$ at $\displaystyle a;$ explain part $\displaystyle (a)$ geometrically by interpreting $\displaystyle f(x), L(x)$ and $\displaystyle e(x)$ graphically.