# Math Help - Linearization of f'(a).

1. ## Linearization of f'(a).

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

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

2. a) The usual proof for this (at least the one from Spivak's Calculus) uses L'Hopital's rule, but going through it I realized I'm not sure if it's valid if we only assume f'(a) exists (and not that the derivative exists in some neighbourhood of a).

b)Just notice that L is the straight line tangent to f at a.