Algebra on Equivalent Infinitesimals Question:

Is algebra on equivalent infinitesimals valid?


Example:
$$\sqrt{1+x} - 1 \sim_0 \frac{x}{2} \longrightarrow \sqrt{1+x} \sim_0 \frac{x}{2} + 1$$
I'm aware that both the left and right equalities are true, but is this just a coincidence that it looks like I added $1$ to both sides of the left equality to get the the right?

We can be more rigorous using remainders:
$$\sqrt{1+x}=1+\frac x2+\mathcal O(x^2)$$
Where $\mathcal O(x^2)$ here means that as $x\to0$, then the remainder is at most $Cx^2$ for some $C$.  Using Taylor polynomials and their remainders, we can see that for $|x|<0.5$,
$$\left|\sqrt{1+x}-1-\frac x2\right|\le2^{-2.5}x^2$$
Indeed, try this out and you will see it holds true, which allows you to show that
$$\sqrt{1+x}\sim_01+\frac x2$$

Anyways, in general, when you write something like
$$\sqrt{1+x}-1\sim_0\frac x2$$
It means that they both have the same growth rates (their difference is bounded by something that goes to zero faster than them) and from there it's easy to show addition and multiplication can be moved around.
