misunderstanding from definition of SNR
In accordance with your calculations, the power of the noise would also be zero.  And 0/0 is indeterminate.  In any case, the SNR is only important for the duration of the signal. That determines the detectability of the signal. SNR has no meaning if there is no signal.Also, by dividing by T, you are calculating average power.  Again, the meaningful SNR is the peak signal divided by the noise or, in some cases, the average power over the duration of the signal divided by the average noise power over the duration of the signal, not to infinity.