Hello,

The limit (when x->0) of |sgn(x)| is 1. This is correct. The limit (when x->0) of xsin(1/x) is 0. This is also correct. However, the composition of these two functions has NO LIMIT when x->0, namely lim (x->0) |sgn(xsin(1/x))| does not exist. However, Mathematica 9.0 and 10.0 fail to detect this, i.e. show the limit as 1:

Limit[Abs[Sign[x*Sin[1/x]]],x->0]
1

If you don't believe me that the limit does not exist, then read page 134 (Example 19) in the book "Mathematical Analysis, vol1" by V.A. Zorich (ISBN 3-540-40386-8 Springer-Verlag Berlin). Here is the relevant part of this page: