1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 - 3x + 2}{x^2 - 4}$$. 2. **Recall the formula and rules:** To find limits involving rational functions, first try direct substitution. If it results in an indeterminate form like $$\frac{0}{0}$$, factor numerator and denominator and simplify. 3. **Evaluate by direct substitution:** Substitute $$x=2$$: $$\frac{2^2 - 3(2) + 2}{2^2 - 4} = \frac{4 - 6 + 2}{4 - 4} = \frac{0}{0}$$ which is indeterminate. 4. **Factor numerator and denominator:** $$x^2 - 3x + 2 = (x - 1)(x - 2)$$ $$x^2 - 4 = (x - 2)(x + 2)$$ 5. **Simplify the expression:** $$\frac{(x - 1)(x - 2)}{(x - 2)(x + 2)} = \frac{x - 1}{x + 2}, \quad x \neq 2$$ 6. **Evaluate the simplified limit:** $$\lim_{x \to 2} \frac{x - 1}{x + 2} = \frac{2 - 1}{2 + 2} = \frac{1}{4}$$ **Final answer:** $$\boxed{\frac{1}{4}}$$