There is an excellent way to understand the limit of a sequence by using a calculator. In scientific calculators that include geometric functions (such as cosine, sine, and tangent), a limit is easy to see: Find x1 = cos (1), then x2 = cos (x1), and so on. Just put the calculator in the “radian” mode, enter “1,” and then hit the “cosine” key repeatedly. The number will start at “0.540302305,” then change to “0.857553215,” and keep changing. As it approaches about the twentieth change, the amount gets closer and closer to a number that begins as “0.73 …”. This indicates the limit of the sequence is being reached.
