A theorem I am given states that a function $f(x) = p_n(x) + O((x-x_0)^{n+1})$ where $p_n(x)$ is the Taylor expansion of $f(x)$ without the remainder error term.
I have a few questions to prove using this theorem and the first two are as follows:
$\sin(x) = o(1)$ as $x \rightarrow 0 $
$\sin(x) = O(x)$ as $x \rightarrow 0 $
This is my first introduction to Landau's notation. I understand, at least I think I do, the concept behind "big o" and "little o" and I have the notes for the notation means. However, I couldn't find any questions that satisfied the use of that theorem.
For the first one I tried taking the Taylor expansion to say and using the theorem I got that $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} + O((x)^6)$ as $x \rightarrow 0$ but I couldn't get any further than that. I also tried taking the other angle and starting with $\sin(x) = o(1) \Rightarrow \sin(x) = O(1)$ which means that $\lvert \sin(x)\rvert \le C \lvert 1\rvert$
Could anyone please try to help me through at least the first one? I'm little lost on exactly what I'm trying to do here.
EDIT: I probably should have clarified that the question is asking that those two relations are proven using the theorem I put at the top of this question.
Thank you in advance.
Hints:
For the first one: saying a function $f$, defined in a neighbourhood of $0$, is $o(1)$ simply means that $\lim_{x\to 0} f(x)=0$.
For the second one, saying that $f(x)=O(x)$ means that $\dfrac{f(x)}{x}=O(1)$, i.e. is bounded in a neighbourhood of $0$ (except at $0$, where it is undefined).