This is a silly question to ask because the answer might be obvious, but I still have my doubts.
If given a circuit composed only of resistors, inductors, and capacitors, if its impulse response decays over time, there should be negative exponents in that response. But that means that the complex roots of the characteristic polynomial should have negative (or zero) real parts. I assumed this was because the characteristic polynomial has positive coefficients but I later found out that not all polynomials with positive coefficients have no positive real parts in their roots. For instance, take a look at \$s^5+s^4+s^3+s^2+s+1\$.
I know there might be some physical explanation to this even though I'm looking from a mathematical standpoint.
Best Answer
In a circuit that is composed of only Ls, Cs and Rs ...
An input impulse will store energy in the Ls and Cs. The stored energy will be dissipated in the Rs, and will tend to zero over time.
Conversely, if there are no Rs, no means of dissipation, then the energy will remain stored, and the impulse response will last indefinitely.