Like Coquet said, the MIDI instrument is responsible for the ability to pitchbend. Most synths and sampler have a default pitchbend range + or - 2 semitones.
Also, if you just enter the notes with your mouse, then the software doesn't have pitch bend information. If you are recording a MIDI sequence and you do pitch bend (assuming the plug in allows for pitch bend), Ableton will record it. If you are just keyboard/mouse entering, then the video above will instruct on how to automate pitch bending. This is also good for other channel automation as well.