1. **Statement of the problem:** We need to find the limit as $x$ approaches 4 of the function: $$\frac{6 - \sqrt{x} + 5\sqrt[3]{x} + 4}{2\sqrt{x} + 5 - 3\sqrt[3]{x} + 4}$$ without using L'Hôpital's rule. 2. **Rewrite the expression:** Simplify the denominator by combining like terms: $$2\sqrt{x} + 5 - 3\sqrt[3]{x} + 4 = 2\sqrt{x} + 9 - 3\sqrt[3]{x}$$ So the expression becomes: $$\frac{6 - \sqrt{x} + 5\sqrt[3]{x} + 4}{2\sqrt{x} + 9 - 3\sqrt[3]{x}} = \frac{10 - \sqrt{x} + 5\sqrt[3]{x}}{2\sqrt{x} + 9 - 3\sqrt[3]{x}}$$ 3. **Evaluate each component at $x=4$:** Calculate $\sqrt{4} = 2$ Calculate $\sqrt[3]{4} = \sqrt[3]{4}$ (approx 1.587) Numerator at $x=4$: $$10 - 2 + 5 \times 1.587 = 10 - 2 + 7.935 = 15.935$$ Denominator at $x=4$: $$2 \times 2 + 9 - 3 \times 1.587 = 4 + 9 - 4.761 = 8.239$$ 4. **Compute the limit:** Since direct substitution does not lead to an indeterminate form, the limit is: $$\lim_{x \to 4} \frac{6 - \sqrt{x} + 5\sqrt[3]{x} + 4}{2\sqrt{x} + 5 - 3\sqrt[3]{x} + 4} = \frac{15.935}{8.239} \approx 1.933$$ 5. **Conclusion:** Therefore, the limit as $x$ approaches 4 is approximately $1.933$.