No, Frank, I am not saying that at all. It is a bit more subtle than that. I am saying that the limit does not exist, because the conditions of Theorem 5 about the limit of a composite function (see page 133 of the book I quoted) are NOT satisfied in this particular case. I would have included the page with the Theorem 5 for your convenience, but it assumes the knowledge of other concepts (like the oscillation of a function, a base etc) and all these would require several pages to be included.
Now, as I said in the second message (the first one in this thread), if a function y = f(x) has a limit y0 (x->x0) and function g = g(y) has a limit g0 (y->y0) it does NOT imply that the composite function g(f(x)) has a limit g0 (x->x0). You need all of the conditions of the Theorem 5 to be satisfied for this to be the case.
Specifically, in the example under consideration, what prevents the conditions of Theorem 5 to be satisfied is the fact that the function Abs[Sign[x*Sin[1/x]]] has countably many zeros in the deleted neighbourhood of the point x=0:
In[1]:= Reduce[Abs[Sign[x*Sin[1/x]]] == 0, x, Reals]
1 1 1
Out[1]= C[1] ∈ Integers && ((C[1] <= -1 && (x == --------- || x == --------------)) || (C[1] >= 1 && x == ---------) ||
2 Pi C[1] Pi + 2 Pi C[1] 2 Pi C[1]
1
> (C[1] >= 0 && x == --------------))
Pi + 2 Pi C[1]
Btw, why does the Subject of this thread have a prefix "Avoid " in it?