How to find the limit of recurrence relation

Hey, first off: I apologise if I've put this in the wrong section. I did a search to see if I could find a topic that could help me with my problem (unfortunately I couldn't find a topic that could satisfy my query) and similar questions seemed to have been posted in this section.

I need to find the limit of this recurrence relation:

In detail, what is the method for finding the limit of this sequence? Is there a general rule/formula for finding the limit of such recurrence relations?

Thanks.