Finding the voltage gain at low frequencies
Well something doesn't seem right here. I haven't checked the equation for \$V_g(s)\$, but starting from there, I can't arrive at the second equation:

We start with:
 $$ V_g(s) = V_i(s)\frac{R_{in}}{R_{in}+R+\frac{1}{sC_{C1}}} $$

Divide by \$V_i(s)\$:

$$ \frac{V_g(s)}{V_i(s)} = \frac{R_{in}}{R_{in}+R+\frac{1}{sC_{C1}}} $$

Pull out the term \$\frac{R_{in}}{R_{in}+R}\$
$$ \frac{V_g(s)}{V_i(s)} = \frac{R_{in}}{R_{in}+R}\frac{1}{1+\frac{1}{sC_{C1} (R_{in}+R)}} $$

Multiply by \$\frac{s}{s}\$:
$$ \frac{V_g(s)}{V_i(s)} = \frac{R_{in}}{R_{in}+R}\frac{s}{s+\frac{1}{C_{C1} (R_{in}+R)}} $$

So either I'm doing it wrong, or the textbook has forgotten to remove the \$s\$ in the second denominator.

(English isn't my first language and I struggle with the names for basic mathematic operations, in German the last step would be "erweitern" and the step before "ausklammern")