I found this approximation made in a paper. The equation was essentially,
$H = \sum_{i=0}^{N-1}\frac{\lambda}{2}((x^2+a^2)^{1/2}-a)^2$
and that for small x this could be approximated as
$H = \sum_{i=0}^{N-1}\frac{\lambda}{8a^2}x^4$
I can't even begin to understand how they got this. I tried a taylor series but did not get anywhere close to this. This was a peer reviewed paper, any ideas how someone could take logical steps to make this approximation?
$$\sqrt{x^2 + a^2} - a = \frac{x^2}{\sqrt{x^2 + a^2} + a} \sim_{x \to 0} \frac{x^2}{2a},$$ where $f(x) \sim_{x \to 0} g(x)$ is shorthand for $\lim_{x \to 0} \frac{f(x)}{g(x)} = 1$.