1. **State the problem:** We need to evaluate the limit $$\lim_{x \to -2} \frac{x^{2} + 4x + 4}{x^{2} - 2x - 8}$$. 2. **Factor numerator and denominator:** - Numerator: $$x^{2} + 4x + 4 = (x + 2)^2$$. - Denominator: $$x^{2} - 2x - 8 = (x - 4)(x + 2)$$. 3. **Simplify the expression:** $$\frac{(x + 2)^2}{(x - 4)(x + 2)} = \frac{x + 2}{x - 4}, \quad x \neq -2$$ (we cancel one factor of $(x+2)$). 4. **Evaluate the limit:** Substitute $x = -2$: $$\frac{-2 + 2}{-2 - 4} = \frac{0}{-6} = 0$$. 5. **Final answer:** The limit is $0$.