For a sequence defined by a formula normally the usual limit rules allows one to find its limit. But for a sequence defined by a recurrence, up to now I have only seen some refined ad hoc methods, mostly in Problems.
The Trick explains that "There is one trick that is (...) first to prove that a limit exists, and then to use the recurrence to determine what the limit must be" illustrated by the example of the recurrence $ a_{n+1}=a_{n}/2+1/a_{n}$, $a_{0}=2$.
Question: Are there relatively general methods to find the limit of a sequence defined by a recurrence?