1. **State the problem:** We want to find the limit as $x$ approaches 4 of the expression $$\frac{6 - \sqrt{x} + 5 \sqrt[3]{x + 4}}{2\sqrt{x} + 5 - 3 \sqrt[3]{x + 4}}.$$\n\n2. **Evaluate the numerator and denominator at $x=4$: ** - Numerator: $6 - \sqrt{4} + 5 \sqrt[3]{4 + 4} = 6 - 2 + 5 \sqrt[3]{8} = 6 - 2 + 5 \times 2 = 4 + 10 = 14$ - Denominator: $2 \sqrt{4} + 5 - 3 \sqrt[3]{4 + 4} = 2\times 2 + 5 - 3 \times 2 = 4 + 5 - 6 = 3$ \n3. **Since the denominator at $x=4$ is $3 \neq 0$ and numerator is $14$, the limit is simply:** $$\lim_{x \to 4} \frac{6 - \sqrt{x} + 5 \sqrt[3]{x + 4}}{2\sqrt{x} + 5 - 3 \sqrt[3]{x + 4}} = \frac{14}{3}.$$