Suppose you discover that the first four terms of the sequence are 1.5, 1.25, 1.125 and 1.0625;  

what is your best guess for its limit, s?

Extrapolation here, like any form of future prediction or clairvoyance requires an assumption about the unknown.

Here you can note that the differences between successive terms are .25, .125 and .0625 which decline by a factor of 2 from term to term. If you assume that this pattern continues, these differences will form a geometric sequence, which you can sum:

1/4+1/8+1/16+...=1/2,

and you would naturally guess that the limit is the first term, 1.5, less the sum of all these differences, 1/2, and hence 1.