1. **Evaluate the limit**: $$\lim_{x \to +\infty} \frac{3 - \sqrt{x}}{3 + \sqrt{x}}$$ 2. **Find partial derivatives** of $$z = 2x^3 + 7x^2y - 3y + 10$$: (a) $$\frac{\partial z}{\partial x}$$ (b) $$\frac{\partial z}{\partial y}$$ 3. **Evaluate integrals**: (a) $$\int \frac{3x + 4}{(x - 1)(x + 1)} \, dx$$ (b) $$\int \sin 10x \cos 7x \, dx$$ 4. **Find volume of revolution** of $$f(x) = 5x^2$$ on $$[0,10]$$ about the x-axis. --- **Step 1: Limit evaluation** 1. The limit is $$\lim_{x \to +\infty} \frac{3 - \sqrt{x}}{3 + \sqrt{x}}$$. 2. Divide numerator and denominator by $$\sqrt{x}$$: $$\frac{\frac{3}{\sqrt{x}} - 1}{\frac{3}{\sqrt{x}} + 1}$$ 3. As $$x \to +\infty$$, $$\frac{3}{\sqrt{x}} \to 0$$, so the limit becomes: $$\frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1$$. **Answer:** $$-1$$. --- **Step 2: Partial derivatives** Given $$z = 2x^3 + 7x^2y - 3y + 10$$. (a) Partial derivative with respect to $$x$$: Use power and product rules, treat $$y$$ as constant: $$\frac{\partial z}{\partial x} = 6x^2 + 14xy$$. (b) Partial derivative with respect to $$y$$: Treat $$x$$ as constant: $$\frac{\partial z}{\partial y} = 7x^2 - 3$$. --- **Step 3: Integrals** (a) $$\int \frac{3x + 4}{(x - 1)(x + 1)} \, dx$$ 1. Factor denominator: $$(x-1)(x+1) = x^2 - 1$$. 2. Use partial fractions: $$\frac{3x + 4}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$$. 3. Multiply both sides by denominator: $$3x + 4 = A(x+1) + B(x-1) = (A+B)x + (A - B)$$. 4. Equate coefficients: - For $$x$$: $$3 = A + B$$ - For constant: $$4 = A - B$$ 5. Solve system: Adding: $$3 + 4 = 2A \Rightarrow A = \frac{7}{2}$$ Subtracting: $$3 - 4 = 2B \Rightarrow B = -\frac{1}{2}$$ 6. Integral becomes: $$\int \left( \frac{7/2}{x-1} - \frac{1/2}{x+1} \right) dx = \frac{7}{2} \int \frac{1}{x-1} dx - \frac{1}{2} \int \frac{1}{x+1} dx$$ 7. Integrate: $$= \frac{7}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C$$. (b) $$\int \sin 10x \cos 7x \, dx$$ 1. Use product-to-sum formula: $$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$ 2. Substitute: $$= \int \frac{1}{2}[\sin(17x) + \sin(3x)] dx = \frac{1}{2} \int \sin 17x \, dx + \frac{1}{2} \int \sin 3x \, dx$$ 3. Integrate: $$= \frac{1}{2} \left(-\frac{\cos 17x}{17}\right) + \frac{1}{2} \left(-\frac{\cos 3x}{3}\right) + C = -\frac{\cos 17x}{34} - \frac{\cos 3x}{6} + C$$. --- **Step 4: Volume of revolution** Find volume when $$f(x) = 5x^2$$ is revolved about x-axis from $$x=0$$ to $$x=10$$. 1. Volume formula: $$V = \pi \int_0^{10} [f(x)]^2 dx = \pi \int_0^{10} (5x^2)^2 dx = \pi \int_0^{10} 25x^4 dx$$ 2. Integrate: $$25 \pi \int_0^{10} x^4 dx = 25 \pi \left[ \frac{x^5}{5} \right]_0^{10} = 25 \pi \times \frac{10^5}{5} = 25 \pi \times 20000 = 500000 \pi$$ **Answer:** $$500000 \pi$$ cubic units.