Disclaimer: I'm an engineer, not a mathematician
Below is the derivation of the voltage across one capacitor with two capacitors in series, so C1 and C2 are greater than zero. $\omega$ is the frequency.
$ V_{C2} = \dfrac{Z_{C2}}{Z_{C1} + Z_{C2}}V1 = \dfrac{\dfrac{1}{j\omega C2}}{\dfrac{1}{j\omega C1} + \dfrac{1}{j\omega C2}} V1 = \dfrac{\dfrac{C1}{j\omega C1 C2}}{\dfrac{C2}{j\omega C1 C2} + \dfrac{C1}{j\omega C1 C2}} V1 = \dfrac{C1}{C1 + C2} V1 $
I understand that the last step is not allowed if $\omega$ = 0, and that in that case I should take the limit:
$ V_{C2} = \displaystyle \lim_{\omega \to 0} \dfrac{\dfrac{C1}{j\omega C1 C2}}{\dfrac{C2}{j\omega C1 C2} + \dfrac{C1}{j\omega C1 C2}} V1 = \dfrac{C1}{C1 + C2} V1 $
That makes sense to me, but I wonder if I didn't skip a step for the limit.
I remember when I was a student I found most steps in a proof "trivial", much to the professor's vexation :-)
So, are there additional steps I shouldn't have skipped?
I'm not a mathematician, so please type slowly :-)