say i have a circuit like this, i understand how to derive the expressions for critically,under and over-damped.
These are the roots of characteristic equation
\(S_1_,_2 = sqrt{\frac{R}{2L}} \stackrel{+}{-} \sqrt{\frac{R}{2L}^2-\frac{1}{LC}} \)

Now my question is, to get expression for LC circuit, we can simply make R=0.

however, what happen if i want to make it RL, or RC? Actually, this is a question that bothers me quite some time, it seems like i just need to get rid of the terms that contain L or C, otherwise the terms will be infinite.

also, if either L or C is missing, there is no resonant frequency?

 

When you make L or C equal to zero the system is not described by a quatradic equation and thus you cannot use the formula you have posted.

If L or C is missing there is not resonant frequency because you need both L and C to resonate at a particular frequency. When you have only an L or C then you have a cut-off frequency.