To evaluate the limit, we can simply substitute the value of x=1 in the expression inside the limit. However, direct substitution yields an indeterminate form of 0/0. Therefore, we need to manipulate the expression before substituting the value of x.
We can factorize the denominator of the expression inside the limit as follows:
x^2 - 3x + 2 = (x-2)(x-1)
So, the expression inside the limit becomes:
\[\frac{1-x}{(x-2)(x-1)}\]
Now, we can simplify the expression by canceling out the common factor of (x-1) in the numerator and denominator:
\[\frac{1-x}{(x-2)(x-1)} = \frac{1}{x-2}\]
Now, we can substitute the value of x=1 in the simplified expression to get the limit value:
\[\lim_{x \to 1}\frac{1}{x-2} = -\infty\]
Therefore, the limit of the given expression as x approaches 1 is negative infinity (-∞), which is not one of the given options.