I've heard that mainly most most mentors give their mentee's their full attenton and never push their mentee's. Mentors help their mentee's make a goal in life and set it, most mentee's learn from their mistakes and make wise decisions. Mentors are always specific about their actions. Mentors are always supportive and encourage their mentee's. Mentors care about their mentee's progress. And last mentors are usually well admired.

Well I think Elijah and Elisha both cared about each other and especially Elijah cared about Elisha because he was Elisha's mentor and cared alot about him and his progress toward his relationship with God and the Bible. 

And being the best prophet he can be in spreading God's word.