Let $a_1$ be the first term of of sequence. Now define
$$a_{n+1}=R(a_n)$$ where R is any recurrence relation. Assuming that $\lim_{n\rightarrow }a_n$ exists , I am looking for a relationship between the limit and the first term of the sequence. Is there any known relationship(as general as possible) between the first term and the limit involving $R$?