Augustus gave up on the idea of making the Elbe River the boundary. Later in the first century the Romans expanded to the east of the Rhine in what is now Bavaria, crating the “Agri Decumates” which gave them a shorter frontier to defend. The local Germans in that area were the Alamanni (or Alemanni). The modern French word for Germany comes from the Alemanni.
Thanks. The Romans cleared forest in Jutland (the trees grew back over the next few centuries for use for shipbuilding) and were at least for a time in control of what is now Copenhagen. In recent years northern and eastern battle sites have been discovered and excavated, at least one large one dating from the Crisis of the 3rd C, a Roman victory, during the short reign of Maximinus Thrax, who somehow doesn't strike me as a great administrator. The written record is bound to be fragmentary, but the archaeology is there.