1. The problem is to find the limit \( \lim_{x \to -2} \sqrt{x^2 + 4x + 4} \). 2. First, recognize that the expression inside the square root is a quadratic: \(x^2 + 4x + 4\). 3. Factor the quadratic: \(x^2 + 4x + 4 = (x + 2)^2\). 4. Substitute the factorization back into the limit expression: \(\lim_{x \to -2} \sqrt{(x + 2)^2} \). 5. The square root of a square is the absolute value: \(\sqrt{(x + 2)^2} = |x + 2|\). 6. So the limit becomes \(\lim_{x \to -2} |x + 2|\). 7. As \(x\) approaches \(-2\), \(x + 2\) approaches 0. 8. Since the absolute value function is continuous, \(\lim_{x \to -2} |x + 2| = |\lim_{x \to -2} (x + 2)| = |0| = 0\). 9. Therefore, the limit is \(0\).