Elisha requests Eliyahu's recognition of him as his successor, although it is clear to both of them that Elisha is different from his master, and is not prepared to negate himself. What he wants is for Eliyahu's spirit to rest upon him in addition to his own, independent spirit. If God opens Elisha's eyes and shows him the vision of Eliyahu's ascent to heaven, alive, with the understanding that Eliyahu's mission is not yet complete, and that his path in this world must be continued, with a softening of the way – then he will be Eliyahu's heir and successor. If Elisha does not merit this, then it will be clear that God does not desire for Elisha to be Eliyahu's successor. It will be clear that the special attributes of this disciple are independent and not a continuation of those of his master. 

Eliyahu and Elisha
Contrast Between Eliyahu and Elisha
Inheritance
Courtesy of the Virtual Beit Midrash, Yeshivat Har Etzion