Trouble understanding the proof of Taylor expansion for one variable functions

Question

In this link I don't understand two last lines in that segment.

\begin{align} \lim_{x\to a} \frac{f(x) - P(x)}{(x-a)^k} &= \lim_{x\to a} \frac{\frac{d}{dx}(f(x) - P(x))}{\frac{d}{dx}(x-a)^k} = \cdots = \lim_{x\to a} \frac{\frac{d^{k-1}}{dx^{k-1}}(f(x) - P(x))}{\frac{d^{k-1}}{dx^{k-1}}(x-a)^k}\\ &=\frac{1}{k!}\lim_{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\ &=\frac{1}{k!}(f^{(k)}(a) - f^{(k)}(a)) = 0 \end{align}

I know it says that it follows from the definition of the derivative at $x=a$ but can someone explain this to me more thoroughly ?

Answer 1

First, remember that $$ \lim_{x\to a} \frac{g(x) - g(a)}{x-a} = g'(a) $$ whenever $g$ is differentiable at $a$. Some people are more familiar with the limit in the form $\lim_{h\to0} \frac{f(a+h) - f(a)}{h}$, but they are equivalent by changing the variables from $h$ to $x=a+h$.

Also, there seems to be a typo in the Wiki article. It says $f^{(j)}(a) = P^{(j)}(a)$ for all $j$ from $1$ to $k-1$, but it also holds for $j=k$.

The rest of it is the old add-and-subtract-the-same-thing trick: $$ \begin{split} \lim_{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a} &= \lim_{x\to a} \frac{f^{(k-1)}(x) -f^{(k-1)}(a) + f^{(k-1)}(a) - P^{(k-1)}(x)}{x-a} \\ &= \lim_{x\to a}\left(\frac{f^{(k-1)}(x) -f^{(k-1)}(a)}{x-a} - \frac{P^{(k-1)}(x) - f^{(k-1)}(a)}{x-a}\right) \\ &= \lim_{x\to a}\frac{f^{(k-1)}(x) -f^{(k-1)}(a)}{x-a} - \lim_{x\to a}\frac{P^{(k-1)}(x) - P^{(k-1)}(a)}{x-a} \\ &= f^{(k)}(a) - P^{(k)}(a) = f^{(k)}(a) - f^{(k)}(a) = 0 \end{split} $$

Answer 2

This follows from L'Hopital Rule applied $k$ times: see here. This isn't just an immediate consequence of the definition of derivative.

Answer 3

$P(x) = \sum_\limits{n=0}^k \frac {f^{(n)}(a)(x-a)^n}{n!}\\ P^{(k-1)}(x) = f^{(k-1)}(a) + f^{(k)}(a)(x-a)$

Plug that into to:

$\frac{1}{k!}\lim_\limits{x\to a} \frac{f^{(k-1)}(x) - P^{(k-1)}(x)}{x-a}\\ \frac{1}{k!}\lim_\limits{x\to a} \frac{f^{(k-1)}(x) - f^{(k-1)}(a) - f^{(k)}(a)(x-a)}{x-a}\\ \lim_\limits\limits{x\to a} \frac{f^{(k-1)}(x) - f^{(k-1)}(a)}{x-a} - \frac{f^{(k)}(a)(x-a)}{x-a}$

And the equation on the left equals $f^{(k)}(a)$ by the definition of the derivative.

What does it mean? $P(x)$ is the best $k$ degree polynomial in a neighborhood of $a.$