Find limit as x approaches 1 for x-1/ x^2-3x+2

I'm going to assume that you meant the following:

  (x-1)

x^2-3x+2

Factoring the denominator, we get:

  (x-1)

(x-1)(x-2)

Notice that we have x-1 in both the numerator and the denominator.  This factor cancels.  HOWEVER:  x can still not equal 1, which would lead to div. by zero.  With the factor x-1 cancelled, we have   1 / (x-2).  As x approaches 1 from either side, y approaches 1 / (1-2), or 1 / (-1), or -1.  Thus, the limit exists and is -1.