I am currently struggling to wrap my head around some basics and would like some clarification if possible. I have been asked to find the order (Big-O) of some non-basic functions. One example being $\log (2+sin(x))$ as $x$ tends to $0$. My attempt at this question has me using the expansion for $\sin x=x-\frac 16 x^3+…$. This gives the function as $\log(2+x-\frac 16 x^3+…)$. Since $x \to 0$, the constant 2 is the dominant term. This leaves us with $\approx$ $\log 2$ which is a constant. Does this mean that the order of the function is $\mathcal{O}( 1)$?
[Math] Finding the order of functions (Big-O)
asymptoticsfunctions
Best Answer
Yes indeed. But remember, you need to specify the point of approximation : You can't just say $$log(2+sin(x)) = \mathcal{O}(1)$$
You should say :
$$log(2+sin(x)) \underset{x\to 0}{=} \mathcal{O}(1)$$
In this particular case it's not relevant but it's a good practice and will help you get better with those notations.