I feel like I do not quite grasp the concept of Big O Notation. From my understanding, if $f(x) = O(g(x))$ then $f(x)$ is at most $g(x)$ multiplied by some constant C, which makes decent sense to me. However, I become very confused when trying to apply the constant.
For my homework in mathematical analysis, my teacher assigned three functions such that $f(x)=O(x^k)$ and told us to find the smallest possible k in each instance. I think I was able to solve the first one but am unable to make any head way on the other two. They are as follows:
- $\sin(x^2)$
- $1-\cos(x^2)$
- $\tan(x^3)$
My problem in understanding these questions is pretty fundamental and I feel that once I get one I will be able to solve the others. Any help is appreciated, but I'm not really looking for plain answers with no explanations.
As stated previously, I think I was able solve the first one and here is my logic behind it:
- I broke $\sin(u)$ into its taylor series, such that: $$sin(u)=u-\frac{u^3}{3!}+\frac{u^5}{5!}+\cdots$$
- I substitued $x^2$ for $u$ such that: $$\sin(x^2)=u-\frac{x^6}{3!}+\frac{x^{10}}{5!}+\cdots+(-1)^n\frac{x^{4n-2}}{(2n-1)!}$$
- From here, I followed a procedure in my book (which could be wrong!) such that $\sin(x^2)=O(x^{4n})$, and, therefore, $4n$ is the smallest value k for the circumstance
Any help, whether it be corrections, hints, or solutions are quite welcome.