Why did Elijah ask Elisha to stay behind in 2 Kings Chapter 2?
The account of Elijah’s preparations to depart and Elisha’s determination to follow presupposes previous revelation, not in Scripture, that this day was to be Elijah's last on earth (v.3). By repeatedly granting Elisha permission to remain behind (v.2, et al.), Elijah was testing Elisha's commitment to himself and to his calling as Elijah's successor (see also Matt. 4:1-11; Luke 22:31-62; John 21:15-25). In a sense, Elijah was giving Elisha the opportunity to decline the difficult life and calling of a prophet.
Elisha's refusal to speak of Elijah's departure, much less to turn back (vs.3, 5), probably reflects Elisha's sorrow at the prospect of losing his friend and mentor, and a commitment to honoring Elijah by serving in his place. Also, it was not uncommon for prophets to give a valuable parting blessing (see also Gen. 49; Deut. 33), and Elisha apparently did not want to miss that.