One can analyze the infinitesimal behavior via the second derivative test and higher-order derivative test, if the function is differentiable enough, and if the first non-vanishing derivative at formula_13 is a continuous function, one can then conclude local behavior (i.e., if formula_104 is the first non-vanishing derivative, and formula_105 is continuous, so formula_106), then one can treat "f" as locally close to a polynomial of degree "k," since it behaves approximately as formula_107 but if the "k"-th derivative is not continuous, one cannot draw such conclusions, and it may behave rather differently.