1. **State the problem:** Find the limit as $x$ approaches 2 of the function $$\frac{x^2 - 3x + 2}{x - 2}.$$\n\n2. **Recall the formula and rules:** The limit of a rational function as $x$ approaches a value where the denominator is zero can sometimes be found by simplifying the expression if the numerator also becomes zero at that point (indeterminate form $\frac{0}{0}$).\n\n3. **Evaluate the function at $x=2$ directly:**\n$$\frac{2^2 - 3\cdot 2 + 2}{2 - 2} = \frac{4 - 6 + 2}{0} = \frac{0}{0},$$ which is indeterminate. So, we simplify the expression.\n\n4. **Factor the numerator:**\n$$x^2 - 3x + 2 = (x - 1)(x - 2).$$\n\n5. **Simplify the expression:**\n$$\frac{(x - 1)(x - 2)}{x - 2} = x - 1, \quad x \neq 2.$$\n\n6. **Find the limit using the simplified expression:**\n$$\lim_{x \to 2} (x - 1) = 2 - 1 = 1.$$\n\n**Final answer:**\n$$\boxed{1}.$$