Why can higher order infinitesimal be omitted

Why can higher order infinitesimal be omitted

If the increment Δ y of the function can be expressed as the sum of the two quantities of the above characteristics, where a Δ x is called the differential corresponding to the increment Δ X of the independent variable, denoted as dy. if the variable y is the function of the variable x, y = f (x), then Δ Y / Δ x = a + O (Δ x) / Δ x is obtained from Δ y = a Δ x + O (Δ x). When Δ x → 0, from the definition of higher order infinitesimal, O (...)

As for the expression "SiNx ~ x when x → 0, so SiNx = x + O (x) when x → 0" in the textbook, what's the meaning of O (x), that is, high-order infinitesimal? I don't understand why high-order infinitesimal is added in this place!

The limit of (sinx-x) / x = SiNx / X-1 is 0, so sinx-x is a higher order infinitesimal of X, which can be expressed as sinx-x = O (x), then SiNx = x + O (x)

Prove that tan5x and 5x are equivalent infinitesimals

The limit of their quotient is equal to 1. Using the definition, we can get that they are equivalent infinitesimal