Let's solve the given limit problem:

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

Step-by-step solution:

1. Factor the denominator:

   The quadratic in the denominator is $x^2 - 3x + 2$. To factor this, we find two numbers that multiply to 2 and add to -3, which are -1 and -2. So: $x^2 - 3x + 2 = (x - 1)(x - 2)$

   Now the expression becomes: $\lim_{x \to 1} \frac{x - 2}{(x - 1)(x - 2)}$

2. Simplify the expression:

   Notice that $x - 2$ is present in both the numerator and denominator. We can cancel the common factor: $\lim_{x \to 1} \frac{1}{x - 1}$

3. Substitute $x = 1$:

   Now we substitute $x = 1$ into the simplified expression: $\frac{1}{1 - 1} = \frac{1}{0}$

   Since division by zero is undefined, this indicates that the limit does not exist.

Conclusion:

The correct answer is: $\boxed{\text{The limit does not exist.}}$

Would you like more details or further explanations?

Here are some related questions:

1. What are the techniques for simplifying limits with indeterminate forms?
2. Can limits involving division by zero ever exist?
3. How do you factor polynomials in limit problems?
4. What is the difference between one-sided and two-sided limits?
5. What are common strategies for solving limits involving infinity?

Tip: When a limit gives a form like $\frac{0}{0}$, try factoring or applying L'Hopital's Rule to resolve the indeterminate form.