Taylor series of degenerate rational function

Question

I have a rational function $f$ which is well-defined on $(0, 1]$, degenerate at zero. I can find the limits as $x \rightarrow 0$ of $f$ and its derivatives, but find that the third and higher derivatives are zero at zero; so the Taylor series around zero is a quadratic. I find this fact strangely disturbing. Am I right to feel this way? Are there any structural implications of a rational function having a truncated Taylor expansion at a degenerate point that would indicate I've made an error?