i try to find the biggest m that $$f(h) = h^{-2}(\sin(1+h)-2\sin(1)+\sin(1-h))+\sin(1)$$ is $\in O(h^m)$ ($h \to 0, h > 0$)
I tried this: $$ f(h) = h^{-2} (((1+h)-\frac{(1+h)^3}{3!}+\frac{(1+h)^5}{5!} \mp ...) -2(1-\frac{1}{3!}+\frac{1}{5!} \mp...) + ((1-h)-\frac{(1-h)^3}{3!}+\frac{(1-h)^5}{5!} \mp ...))+(1-\frac{1}{3!}+\frac{1}{5!} \mp...) =h^{-2}(2(1-h)-\frac{(1+h)^3-(1-h)^3+2}{3!} + \frac{(1+h)^5-(1-h)^5-2}{5!} \mp ...)+\sin(1)$$ But what now?
Can you help me?
Thanks!
Hint: Use the Taylor approximation around $1$,
$$f(1+h) = f(1) + f'(1)\cdot h + \frac{f''(1)}{2}h^2 + \frac{f'''(1)}{3!}h^3 + \dotsb$$
Use the similar approximation of $f(1-h)$ to obtain
$$f(1+h) + f(1-h) = 2\cdot f(1) + f''(1)\cdot h^2 + \dotsb$$
Use approximations of high enough order to achieve your goal.