1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x-2}{x^2 - 4}$$. 2. **Recall the formula and rules:** When evaluating limits that result in an indeterminate form like $$\frac{0}{0}$$, we try to simplify the expression by factoring or canceling common factors. 3. **Evaluate the expression at $x=2$: $$\frac{2-2}{2^2 - 4} = \frac{0}{4-4} = \frac{0}{0}$$, which is indeterminate. So, we simplify the expression.** 4. **Factor the denominator:** $$x^2 - 4 = (x-2)(x+2)$$. 5. **Rewrite the limit:** $$\lim_{x \to 2} \frac{x-2}{(x-2)(x+2)}$$ 6. **Cancel the common factor $(x-2)$:** $$\lim_{x \to 2} \frac{\cancel{x-2}}{\cancel{x-2}(x+2)} = \lim_{x \to 2} \frac{1}{x+2}$$ 7. **Evaluate the simplified limit:** $$\frac{1}{2+2} = \frac{1}{4}$$ **Final answer:** $$\boxed{\frac{1}{4}}$$