Frequency response of an ideal LC circuit
Does the frequency response of an ideal LC circuit exist?  


Yes, why not?


  Am I wrong?


Yes. It's obvious that the pulse response is a pure harmonic function because it is a resonance circuit without resistance.  

Let the pulse be a voltage dirac pulse of value \$1 V\$ at \$t=0\$, and the output signal the voltage over the capacitor. Then it's clear that at \$t=0\$, \$U_C=1 V\$ and that the impulse response is a harmonic voltage starting at \$U_C = 1 V\$ with an angular frequency \$\omega_0 = \frac{1}{\sqrt{L C}}\$ (IIRC).  

Then \$h(t)=cos(\omega_0 t)\$

Why bothering you with that?
Well, because \$\int_\infty^\infty cos(\omega_0 t) e^{-j \omega t}dt = \frac{1}{2}[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]\$.
I guess that integrating from 0 to \$\infty\$ gives only the \$\delta(\omega-\omega_0)\$ term.