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.
edit:
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
Well, it looks like a bandpass filter. It allows frequencies between a certain range, depending on the values of your circuit elements. It becomes more apparent when you find the transfer function.
If you change those circuit elements into the s-domain, and then write a node equation at [;V_o;], you get:
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.