This problem asks for the value of the limit:

$\lim_{x \to 2} \frac{x^2 - 3x + 2}{x - 1}.$

Step 1: Simplify the numerator

The numerator $x^2 - 3x + 2$ is a quadratic expression that can be factored: $x^2 - 3x + 2 = (x - 1)(x - 2).$

Step 2: Rewrite the expression

Substituting the factored form of the numerator into the original fraction: $\frac{x^2 - 3x + 2}{x - 1} = \frac{(x - 1)(x - 2)}{x - 1}.$

Step 3: Cancel common factors

As long as $x \neq 1$, we can cancel $x - 1$ from the numerator and denominator: $\frac{(x - 1)(x - 2)}{x - 1} = x - 2.$

Step 4: Evaluate the limit as $x \to 2$

Now, substitute $x = 2$ into the simplified expression: $x - 2 = 2 - 2 = 0.$

Final Answer:

$\lim_{x \to 2} \frac{x^2 - 3x + 2}{x - 1} = 0.$

Would you like further details or clarifications?

Here are 5 related questions to deepen understanding:

1. What is the general process for factoring quadratic expressions?
2. How do we determine when a factor can be canceled in a rational function?
3. Why is it important to verify the condition $x \neq 1$ in this problem?
4. Can this approach be extended to higher-degree polynomials?
5. What happens if the numerator cannot be factored easily?

Tip: Always check for common factors between the numerator and denominator when simplifying limits involving polynomials.