Quick question on an approximation from physics I've seen written in a physics problem sheet that for a small $\epsilon>0$, the following approximation is considered
$$
(1+2\epsilon)^{-\frac12}\approx 1-\epsilon.
$$
Any reason why this is the case? Is this any special approximation in physics?

This and many other cases are covered by Newton's Generalized Binomial Theorem. Here, we have $$(1+2\epsilon)^{-\frac12} = \displaystyle\sum_{i=0}^\infty \binom {-\frac 12}i (2\epsilon)^i$$
And of course, $\epsilon^i~\forall~i>0$ is very small in comparison to the rest of the sum so we omit those terms. So, our sum approximates to $$\binom{-\frac 12}0 + \binom{-\frac 12}1(2\epsilon)$$ $$= \boxed{1-\epsilon}$$
