TITLE: Methods to find the limit of a sequence defined by a recurrence
QUESTION [12 upvotes]: 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?

REPLY [11 votes]: Not really. Even once you've shown that a limit exists, a limit of, say, $x_{n+1} = f(x_n)$ is the same as a fixed point of $f$, or a solution to $f(x) - x = 0$. Needless to say it does not make much sense to ask whether there are general methods to find solutions to equations. For example, one can write down solutions to differential equations as fixed points, and there are no general methods to solve differential equations.
In that sense all methods to find limits of recurrences are "ad hoc" in the same way that all methods to solve differential equations or Diophantine equations are "ad hoc."  We do what we can.