Limits of Taylor Series: Is $\sin x=x+O(x^2)$ Correct?

[LagrangeEuler]

We sometimes write that
$$\sin x=x+O(x^3)$$
that is correct if
$$\lim_{x \to 0}\frac{\sin x-x}{x^3}$$
is bounded. However is it fine that to write
$$\sin x=x+O(x^2)$$?

 

[fresh_42]

Yes, but you give away information. You could even say $\sin x = O(x).$

 

[mathman]

Writing $O(x^3)$ says there is no $x^2$ term.