We say that two functions $f,g: \Bbb{R} \to \Bbb{R}$ are equal at order $n$ at the point $a$ when $$\lim_{h \to 0}\frac{f(a+h)-g(a+h)}{h^n} = 0.$$
(a) Show that $f$ is differetiable at $a$ iff there is a function $g$ of the form $g(x) = a_0 + a_1(x-a)$ such that $f,g$ are equal at the first order at $a$.
(b) When there is $f'(a),...,f^{(n)}(a)$ its verified that the function $g$, defined by $$g(x) = \sum_0^{n}\frac{f^{(i)}(a)}{i!}(x-a)^i,$$ is equal to $f$ at the order $n$ at the point $a$.
For the item (b), I don't know if I'm right, but I showed that $$\lim_{x \to a}\frac{f^{(n)}(x) - f^{(n)}(a)}{n!}$$ by using the l'Hospital Rule. But at this point, I cannot proceed since $f^{(n)}$ may not be continuous.
This is a problem from it Calculus on Manifolds by Spivak. The author suggests to calculate $$\lim_{x \to a}\frac{f(x) - \sum_0^{n-1} \frac{f^{(i)}(a)}{i!}(x-a)^i}{(x-a)^n}$$ but I cannot see why it helps.
Also, I would like to add that the book does not talk about Taylor's expansion, or anything related. So far, were presented only very basic results of differentiation.