We address the quality of electrical contact between carbon nanotubes and metallic electrodes by performing first-principles calculations for the electron transmission through ideal 2- and 3-terminal junctions, thus revealing the physical limit of tube-metal conduction. The structural model constructed involves surrounding the tube by the metal atoms of the electrode as in most experiments; we consider metallic (5,5) and n-doped semiconducting (10,0) tubes surrounded by Au or Pd. In the case of metallic tubes, the contact conductance is shown to approach the ideal 4e2/h in the limit of large contact area. For three-terminals, the division of flux among the different transmission channels depends strongly on the metal material. A Pd electrode has nearly perfect tube-electrode transmission and therefore turns off the straight transport along the tube. Our results are in good agreement with some recent experimental reports and clarify a fundamental discrepancy between theory and experiment.