How to solve asymptotic expansion: $\sqrt{1-2x+x^2+o(x^3)}$ Determinate the best asymptotic expansion for $x \to 0$ for:
$$\sqrt{1-2x+x^2+o(x^3)}$$
How should I procede?
In other exercise I never had the $o(x^3)$ in the equation but was the maximum order to consider.

Hint. One may recall the Taylor series expansion, as $u \to 0$,
$$
\sqrt{1+u}=1+\frac{u}2-\frac{u^2}8+\frac{u^3}{16}+o(u^3).
$$ Setting, as $x \to 0$,
$$
u=-2x+x^2+o(x^3)
$$ it gives
$$
1+\frac{-2x+x^2+o(x^3)}2-\frac{(-2x+x^2+o(x^3))^2}8+\frac{(-2x)^3}{16}+o(x^3)
$$ or
$$
1+\frac{-2x+x^2}2-\frac{4x^2-4x^3}8+\frac{-8x^3}{16}+o(x^3)
$$thus, as $x \to 0$,

$$
\sqrt{1-2x+x^2+o(x^3)}=1-x+o(x^3).
$$

