if vi were a dc voltage source, it would have the lowest frequency possible (0 Hz). How do inductors and capacitors behave when DC voltage is applied to them as you look at t -> infinity? What would the voltage across the parallel combination of an inductor and a capacitor have to be if they behave that way?
As freq -> infinity, inductors and capacitors behave oppositely to how they behave for f = 0. Apply the same reasoning as above except with the new simplifications for when f -> infinity.
Oh yeah. I guess your first step would be understanding what the different types of filters are:
if you find that the circuit has:
nonzero 0Hz and zero inf. Hz response, it is probably a lowpass
zero 0Hz and nonzero inf. Hz response, it is probably highpass
zero for both, probably bandpass
nonzero for both, probably bandstop
For analysis, if you change it into the frequency domain ([;s = j*\omega;]), set some values for our elements and vary the frequency, we can see what will happen. I find it rather easy in MATLAB. There are equations to see where your range will be, but I don't know them off the top of my head. Something about 3dB. This is where you can get into design.
I guess your best bet is to do a frequency response and graph the frequencies from like 0Hz to 1MHz. You'll see your bandpass
Also, we can see as we increase R, the output will decrease. So I think it's safe to say that increasing R will increase your damping factor.