[xyzzyz]

The statement about sine above is not something mathematicians would write. It makes little sense to use big-O notation in this context, as it doesn't say anything useful here: the O(x^7) element absolutely dominates the remaining explicit elements of lower order, so including them tells us absolutely nothing. In fact, sin(x) = O(1).

However, mathematicians do indeed use similar notation in this context, that is, little-o notation. It is in fact true that

sin(x) = x -x^3/6 + x^5/120 + o(x^5), x -> 0.

[proto-n]

Have you considered that higher orders of x are in fact smaller when x is near 0?

The parent comment was right and you are wrong, around zero x^5 absolutely dominates x^7 and the big-O notation is used. See for example here [1]

[1] https://en.wikipedia.org/wiki/Taylor_series#First_example

[gizmo686]

https://en.wikipedia.org/wiki/Taylor_series#First_example

https://www.wolframalpha.com/input/?i=taylor+series+sin+x

Notice that in your example, you have o(x^5) and an explicit x^5 term. In my example I have O(x^7), but no explicit x^7 term. It is true that I cannot think of a circumstance where you want to do this abuse of notation and would care if you were forced to use little-o or big-O instead of the other.

In my experience, it happens to be more common to use big-O.

[j2kun]

I have definitely used big-O as the parent described. I think many mathematicians would write it in that way.