How can I do the Taylor equivalent of $\sqrt{1+x^2}$ with perturbation/asymptotic expansion using $\epsilon$?

Question

How can I do the Taylor equivalent of $\sqrt{1+x^2}$ with perturbation/asymptotic expansion using $\epsilon$?

I found some answers, but not very useful.

Expected solution would be similar to the Taylor expansion:

1 + x^2/2 - x^4/8 + O(x^5)