##
Setting

The (sub-)circuit in question looks like this:

(Both OP amps can be considered as ideal.)

^{ – Schematic created using CircuitLab}

It's part of a larger question about an Ackerberg-Mossberg biquad (which looks like this) where one is asked to calculate

a) this subcircuits DC voltage gain \$A_{v,DC}=\frac{V_{out}}{V_{in}}\$, and later

b) its transfer function \$A_v(s)=\frac{V_{out}(s)}{V_{in}(s)}\$.

As I like to derive my answers for DC analysis from the transfer function \$A_v(s)\$, I tried the following:

1) Directly determine the transfer function \$A_v(s)\$, which answers b): $$A_v(s)=\frac{V_{out}(s)}{V_{in}(s)}=\frac{1}{sR_1C_1}$$

It should be the transfer function of a **non-inverting integrator amplifier**.

2) Compute the DC gain by using \$\lim\limits_{s \rightarrow 0}{A_v(s)} \$. That is $$A_{v,DC}=\lim\limits_{s \rightarrow 0}{\left(A_v(s)\right)}=\lim\limits_{s \rightarrow 0}{\left(\frac{1}{sR_1C_1}\right)=\infty}$$

But here's the thing: the solution of a) simply says that \$A_{v,DC}=-\infty\$ which contradicts my answer.

##
Questions

So here's my questions for you:

i) Which answer to a) is correct?

ii) If my answer is wrong, one possible explanation would be that I can't simply use the limit for \$s\rightarrow 0\$ because in reality it's defined as \$s:= \sigma + i\omega\$. This means that I would have to assume \$\sigma=0\$ and do the limit for \$\omega\rightarrow 0\$ which brings me to $$A_{v,DC}=\lim\limits_{\omega \rightarrow 0}{\left(A_v(s=0+i\omega)\right)}=\lim\limits_{\omega \rightarrow 0}{\left|\frac{1}{(0+i\omega)R_1C_1}\right|}=\lim\limits_{\omega \rightarrow 0}{\left|\frac{-i}{\omega R_1C_1}\right|}=-\infty$$ Would this be a correct derivation?

iii) If the above doesn't work, why is it so? And is there a way to calculate a) from b)?

DC analysis can become very confusing when one has to deal with open loop amplifiers, especially in bigger circuits. So it would be very nice if one can derive the DC analysis results from the laplace domain transfer function.

iv) To go beyond the exercise question: I'm curious about the bode plot, because the single pole at \$s=0\$ (and no zeros) suggests that the magnitude goes from infinity to zero (with -20dB/decade) as the frequency goes upwards but what happens with the phase?