Lossy transmission of Gaussian sources over energy limited Gaussian point-to-point and broadcast channels is studied under the infinite bandwidth regime, i.e., when the number of channel uses is unlimited. Using previously known asymptotic achievability and converse results, the energy-distortion exponent, defined as the rate of decay of the square-error distortion as the available energy-to-noise ratio increases without bound, is completely characterized for both the point-to-point and broadcast channel cases. Turning then to the scenario of zero delay transmission, where outage events with arbitrarily small probability are allowed, it is shown that the same energyd istortion exponent as in the infinite-delay case can be achieved in all the studied scenarios.