For values of x not equal to 1, we have (x^2-3x+2)/(x^2-2x+1) = (x-2)(x-1)/(x-1)^2 = (x-2)/(x-1). This has no limit as x tends to 1: more precisely, as x tends to 1 from below, written x -> 1-, it tends to +infinity (that is, increases indefinitely) and as x tends to 1 from above, x -> 1+, it tends to -infinity (increases indefinitely in magnitude but with negative sign).
Given a function N/D, if each of N and D tend to a limit as x->a, say n and d respectively, then you can say: if d is non-zero, N/D -> n/d; if d=0 but n non-zero then N/D does not tend to a limit; if d=0 and n=0 then you can't tell immediately.
There's a useful further rule in the case when N and D both tend to zero: L'Hopital's rule. This says that if N and D are differentiable, N and D both -> 0 and the derivatives N' and D' tend to limits, then N/D has the same limiting behaviour as N'/D'. In your case N' is 2x-3 and D' is 2x-1. As x->1, we have N'->-1, D'->0 showing that there is no limit for N'/D' and so no limit for N/D.