1. **State the problem:** Find the limit $ \lim_{x \to 3} \frac{x^2 - 9}{x - 3} $. 2. **Recall the formula and rules:** When direct substitution results in an indeterminate form $ \frac{0}{0} $, factorization or algebraic simplification is used. 3. **Evaluate direct substitution:** Substitute $ x = 3 $: $$\frac{3^2 - 9}{3 - 3} = \frac{9 - 9}{0} = \frac{0}{0}$$ which is indeterminate. 4. **Factor the numerator:** $$x^2 - 9 = (x - 3)(x + 3)$$ 5. **Rewrite the limit expression:** $$\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3}$$ 6. **Cancel common factors:** $$\lim_{x \to 3} \frac{\cancel{(x - 3)}(x + 3)}{\cancel{x - 3}} = \lim_{x \to 3} (x + 3)$$ 7. **Evaluate the simplified limit:** $$x + 3 \bigg|_{x=3} = 3 + 3 = 6$$ **Final answer:** $$\boxed{6}$$