A question about the argument part of impedance
The impedance Z = V/I changing by time, even goes to infinity
  sometimes; yet we associate the reactance with what??


The impedance \$Z\$ of a circuit element is the ratio of the phasor voltage across and phasor current through, not the ratio of the time domain voltage and current.

So, for example,

$$v_C(t) = A\sin(\omega t) = \Re\{-jAe^{j\omega t}\}$$
$$i_C(t) = \omega C\; A\cos(\omega t) = \Re\{\omega C\;Ae^{j\omega t}\}$$

The phasor representation of the voltage and current are just the (constant) complex numbers multiplying \$e^{j \omega t}\$ thus, the phasor voltage and currents are

$$V_c = -jA$$
$$I_c = \omega C A $$

The impedance of the capacitor is then

$$Z_c = \frac{V_c}{I_c} = \frac{-jA}{\omega C A} = \frac{1}{j\omega C}$$

and so the reactance, the imaginary part of the impedance, of the capacitor is

$$X_c = -\frac{1}{\omega C} $$