Frequency response 4

The resistor with the capacitor are a voltage divider. Let's call $Z_c$ the capacitor's impedance and $R$ the resistor's impedance, we can calculate the voltage across the capacitor as

$V_o = V_i \cdot \left(\frac{Z_c}{R + Z_c}\right)$

We can calculate $Z_c$ in function of the frequency as $Z_c = \frac{-j}{2\pi f C}$ with $j = \sqrt{-1}$ . It's easy to see that the capacitor's impedance will go to infinity as the frequency goes to zero, which means that

$\lim_{Z_c \to \infty}\left(\frac{Z_c}{R + Z_c}\right) = \frac{0}{R} = 0$ so $V_o = 0$