1. **State the problem:** We are given an inequality for a function $f(x)$ such that $$4x - 9 \leq f(x) \leq x^2 - 4x + 7$$ for $x \geq 0$. We need to find $$\lim_{x \to 4} f(x).$$ 2. **Recall the Squeeze Theorem:** If $$g(x) \leq f(x) \leq h(x)$$ and $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L,$$ then $$\lim_{x \to a} f(x) = L.$$ This theorem helps us find limits of functions trapped between two others. 3. **Identify the bounding functions:** Here, $$g(x) = 4x - 9$$ and $$h(x) = x^2 - 4x + 7.$$ 4. **Calculate the limits of the bounding functions as $x \to 4$:** $$\lim_{x \to 4} g(x) = \lim_{x \to 4} (4x - 9) = 4(4) - 9 = 16 - 9 = 7.$$ $$\lim_{x \to 4} h(x) = \lim_{x \to 4} (x^2 - 4x + 7) = 4^2 - 4(4) + 7 = 16 - 16 + 7 = 7.$$ 5. **Apply the Squeeze Theorem:** Since both bounding functions approach 7 as $x$ approaches 4, and $f(x)$ is squeezed between them, we conclude: $$\lim_{x \to 4} f(x) = 7.$$