After Sita said strong unbearable words to Lakshmana, he was shattered and prepared to depart in the direction from where 'Ha Lakshman’ sound had come in voice of Rama. He left Sita in the shelter of Vana-devata. Let the forest, creatures save Sita. He departed and Ravana arrived. Jatayu used to protect the hut of Rama. They had met earlier and exchanged how Jatayu and Dasaratha were sweet friends.
Jatayu was not any ordinary vulture. He had participated in deva-asura sangram with Dasaratha, against demons. But, now was an old bird. After Lakshmana left saying these words, it became duty of Jatayu to save Sita.
After Sita was kidnapped and shouted for help, Jatayu attacked Ravana. He fought bravely but Ravana had invincible sword of Shiva. He used it on the old bird and Jatayu fell on the ground.
Like an ideal father-in-law and ideal devotee of Rama and also as per request of Lakshmana, he fought bravely with Ravana.
What if he had followed? Lanka was not easy to enter and once entered, its impossible to stay alive. At this old age, even crossing such a big ocean of 100 Yojana was also not possible for him. With his information only, Rama could know that Sita has been kidnapped by Ravana.
End of Jatayu:
Jatayu left his body in the laps of Rama, head experiencing the loving hand touch of Rama and with his eyes he was staring the dark-bluish divine beauty. The final rites of his body was done by Rama himself.
An interesting commentary of Govindaraaja:
Sugreeva with his army was sitting on Mountains. They saw a lady being forcefully kidnapped and crying for help. They didn't help her. It's better to die for a good cause than to live a life of a coward and unrighteous. Neither did Baali did any favour. He was well capable to saving Sita as he had defeated Ravana in past. But, he remained quite and thus helped Ravana in his adharma.
Rama killed Bali hiding behind the trees the second time Sugreeva went to fight with him. If you are helping adharma by backdoor, you have no right to claim that you were killed by adharma.
Rama didn't save Sugreeva the first time. He was beaten brutally by Bali. Do you think Rama was unable to identify him? One who is being beaten in Sugreeva and stronger one is Bali. There are many other factors that Rama could have easily differentiated between both. Next day, Sugreeva approached Rama through Lakshmana and he was saved by Rama.
