- #1

- 46

- 0

## Homework Statement

Show that if [itex]f^{(n)}(x_0)[/itex] and [itex]g^{(n)}(x_0) [/itex] exist and

[itex] \lim_{x \rightarrow x_0} \frac{f(x)-g(x)}{(x-x_0)^n} = 0 [/itex] then

[itex]f^{(r)}(x_0) = g^{(r)}(x_0), 0 \leq r \leq n [/itex].

## Homework Equations

If f is differentiable then [itex] \lim_{x \rightarrow x_0}\frac{f(x)-T_n(x)}{(x-x_0)^n}=0 [/itex], where T

_{n}is the nth Taylor polynomial.

## The Attempt at a Solution

I'm stuck on how to start the proof at all. I tried induction on r, but didn't get very far with that since I had trouble showing that [itex]f(x_0) = g(x_0)[/itex]. Any ideas on what direction to go to get started?Thanks.

Last edited by a moderator: