1. **Evaluate the limit**: $$\lim_{x \to +\infty} \frac{3 - \sqrt{x}}{3 + \sqrt{x}}$$ - When $x$ tends to infinity, $\sqrt{x}$ also tends to infinity. - Divide numerator and denominator by $\sqrt{x}$ to simplify: $$\frac{\frac{3}{\sqrt{x}} - 1}{\frac{3}{\sqrt{x}} + 1}$$ - As $x \to +\infty$, $\frac{3}{\sqrt{x}} \to 0$, so the limit becomes: $$\frac{0 - 1}{0 + 1} = \frac{-1}{1} = -1$$ 2. **Find partial derivatives of** $z = 2x^3 + 7x^2y - 3y + 10$ (a) Partial derivative with respect to $x$: - Treat $y$ as constant. - $$f_x = \frac{\partial z}{\partial x} = 6x^2 + 14xy$$ (b) Partial derivative with respect to $y$: - Treat $x$ as constant. - $$f_y = \frac{\partial z}{\partial y} = 7x^2 - 3$$ 3. **Evaluate integrals:** (a) $$\int \frac{3x + 4}{(x-1)(x+1)} dx$$ - Factor denominator: $(x-1)(x+1) = x^2 - 1$ - Use partial fractions: $$\frac{3x + 4}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$$ - Multiply both sides by $(x-1)(x+1)$: $$3x + 4 = A(x+1) + B(x-1)$$ - Equate coefficients: - For $x$: $3 = A + B$ - For constants: $4 = A - B$ - Solve system: - Adding: $3 + 4 = 2A \Rightarrow A = \frac{7}{2}$ - Then $B = 3 - A = 3 - \frac{7}{2} = -\frac{1}{2}$ - Integral becomes: $$\int \frac{7/2}{x-1} dx + \int \frac{-1/2}{x+1} dx = \frac{7}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C$$ (b) $$\int \sin 10x \cos 7x dx$$ - Use product-to-sum formula: $$\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$$ - So: $$\int \sin 10x \cos 7x dx = \frac{1}{2} \int [\sin(17x) + \sin(3x)] dx$$ - Integrate: $$= \frac{1}{2} \left(-\frac{\cos 17x}{17} - \frac{\cos 3x}{3}\right) + C = -\frac{\cos 17x}{34} - \frac{\cos 3x}{6} + C$$ 4. **Find volume of revolution of** $f(x) = 5x^2$ **about x-axis from** $x=0$ **to** $x=10$ - Volume formula: $$V = \pi \int_0^{10} [f(x)]^2 dx = \pi \int_0^{10} (5x^2)^2 dx = \pi \int_0^{10} 25x^4 dx$$ - Integrate: $$25 \pi \int_0^{10} x^4 dx = 25 \pi \left[ \frac{x^5}{5} \right]_0^{10} = 25 \pi \times \frac{10^5}{5} = 25 \pi \times 20000 = 500000 \pi$$ **Final answers:** 1. Limit = $-1$ 2. (a) $f_x = 6x^2 + 14xy$ (b) $f_y = 7x^2 - 3$ 3. (a) $\frac{7}{2} \ln|x-1| - \frac{1}{2} \ln|x+1| + C$ (b) $-\frac{\cos 17x}{34} - \frac{\cos 3x}{6} + C$ 4. Volume = $500000 \pi$