Analogue filter analysis (band-pass) The transfer-function:
  $$G(s) = \frac{\beta s}{s^2 + \beta s + \omega_0^2}$$
  
  is to be used in an application that requires the magnitude of the frequency response to be of the band-pass form. 
  
  
  Show that this property is present by considering the cases when $\omega = 0, \omega = 1$ and show also that $\lvert G(j\omega_0)\rvert = 1$. 
  Determine also the equation that defines the frequencies where the filter magnitude is $−3\textrm{ dB}$.
  


I use $G(s)=G(j\omega)|_{s=j\omega}$, then 
$$
\lvert G(j\omega_0)\rvert= \frac{j\beta \omega_0}{-\omega_0^2 + j\beta\omega_0 + \omega_0^2}=1
$$
when $\omega=0$, the numerator equal to $0$ so $\omega=0$ is be filtered out. 

My questions are: 


Could I use the $G(s)=\lvert G(j\omega)\rvert$ directly or $G(s)=G(j\omega)=\lvert G(j\omega)\rvert e^{-jk\omega}$?
I have no idea about the case when $\omega=+\infty$?
The last question, I use $-20\log\lvert G(j\omega)\rvert=-3\textrm{ dB}$, so we can solve the $\omega$ from this equation, is that correct?
Even though this may not be homework, it is a basic textbook problem, and for this reason I just give you a few hints, which should nevertheless help you solve the problem if you put in a bit of effort.


All questions refer to the filter's frequency response, which is simply obtained by evaluating $G(s)$ for $s=j\omega$.
The first question should read "Show that this property ... $\omega=0$, $\omega={\infty}$" (not $\omega=1$).
Your expression for $|G(j\omega)|$ is wrong; you wrote down $G(j\omega_0)$ (without magnitude). Of course, if $G(j\omega)=0$, or $G(j\omega)=1$, then also $|G(j\omega)|=0$ and $|G(j\omega)|=1$, respectively.
In order to see what happens for $\omega\rightarrow\infty$, write down $|G(j\omega)|^2$ (that's maybe easier than the magnitude), and take the limit.
For the 3 dB cut-off frequencies (there must be two positive solutions), use $|G(j\omega)|^2$ derived above and solve $|G(j\omega_c)|^2=\frac12$. Substitute $x=\omega^2$ and solve the resulting quadratic equation for $x$. From that solution, derive the two positive solutions for $\omega_c$.


All of this involves basic math skills, such as computing the (squared) magnitude of a complex number, and solving quadratic equations. If you find that difficult, read up on these basics because you'll need them all of the time when doing signal processing.