Methods to find the limit of a sequence defined by a recurrence

Solution 1:

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.