determining the order of expressions, eg: $\sqrt{\varepsilon(1-\varepsilon)}$

Question

I have to determine the order of loads of expressions as $\varepsilon \to 0$.

Can you help me by giving me an example of how to find the order of $\sqrt{\varepsilon(1-\varepsilon)}$.

Answer 1

The binomial theorem is applicable to arbitrary (even complex) exponents.

$$(1-x)^{r} = 1 - rx + \frac{r(r-1)}{2} x^2 - \frac{r(r-1)(r-2)}{6}x^3 + \dots $$

This series converges absolutely for any $r$, when $|x| \lt 1$.

You can use that, setting $r = \frac{1}{2}$.

See: Binomial Series.

In general, you can try getting the Taylor series at $0$ and see if that works. The above is nothing but the Taylor series of $(1-x)^r$.

Answer 2

Another approach is to use the big O notation and its "absorbing" properties, because then you can keep track of the orders: $$\sqrt{\epsilon\left(1-\epsilon\right)}=\sqrt{\epsilon-O(\epsilon^2)}=\sqrt{\epsilon+O(\epsilon^2)}=\sqrt{\epsilon}\left(\sqrt{1+O(\epsilon}\right)=\sqrt{\epsilon}\left(1+\frac{1}{2}O(\epsilon)\right)=\sqrt{\epsilon}+O(\epsilon^{3/2})$$