1. **Compute** $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$. Factor numerator: $x^2 - 4 = (x-2)(x+2)$. Cancel $(x-2)$: limit becomes $\lim_{x \to 2} (x+2) = 4$. 2. **Compute** $\lim_{x \to 0} \frac{6^{5x} - 1}{x}$. Use derivative definition of $a^x$ at 0: $\lim_{x \to 0} \frac{a^x - 1}{x} = \ln(a)$. So, $\lim_{x \to 0} \frac{6^{5x} - 1}{x} = 5 \ln(6)$. 3. **Compute** $\lim_{x \to 4} \frac{x^3 - 4^3}{x - 4}$. Factor numerator: $x^3 - 64 = (x-4)(x^2 + 4x + 16)$. Cancel $(x-4)$: limit is $\lim_{x \to 4} (x^2 + 4x + 16) = 16 + 16 + 16 = 48$. 4. **Compute** $\lim_{x \to 0} \frac{e^{7x} - 1}{x}$. Using derivative of $e^{kx}$ at 0: limit is $7$. 5. **Compute** $\lim_{x \to 1} \frac{x^3 - 1}{x - 1}$. Factor numerator: $x^3 - 1 = (x-1)(x^2 + x + 1)$. Cancel $(x-1)$: limit is $1 + 1 + 1 = 3$. 6. **Find** $\frac{d^2}{dx^2} (x^4 - 2x + 1)$. First derivative: $4x^3 - 2$. Second derivative: $12x^2$. 7. **Find** $\frac{d}{dx} (x^7 \sin x)$. Use product rule: $7x^6 \sin x + x^7 \cos x$. 8. **Find** $\frac{d}{dx} (x^5 e^x)$. Product rule: $5x^4 e^x + x^5 e^x = e^x (5x^4 + x^5)$. 9. **Find** $\frac{d}{dx} \left( \frac{2x^2 + 3}{e^x} \right)$. Rewrite as $(2x^2 + 3) e^{-x}$. Derivative: $\frac{d}{dx} (2x^2 + 3) e^{-x} + (2x^2 + 3) \frac{d}{dx} e^{-x}$ $= 4x e^{-x} + (2x^2 + 3)(-e^{-x}) = e^{-x} (4x - 2x^2 - 3)$. 10. **Find** $\frac{d^2}{dx^2} (x^2 + 3x + 2)$. First derivative: $2x + 3$. Second derivative: $2$. 11. **Find** $\frac{d^2}{dx^2} (x^8 + x^6 + x^4)$. First derivative: $8x^7 + 6x^5 + 4x^3$. Second derivative: $56x^6 + 30x^4 + 12x^2$. 12. **Find equation of straight lines:** (a) Points (1,2) and (3,4): slope $m = \frac{4-2}{3-1} = 1$. Equation: $y - 2 = 1(x - 1) \Rightarrow y = x + 1$. (b) Points (2,3) and (3,2): slope $m = \frac{2-3}{3-2} = -1$. Equation: $y - 3 = -1(x - 2) \Rightarrow y = -x + 5$. (c) Points (1,1) and (3,3): slope $m = \frac{3-1}{3-1} = 1$. Equation: $y - 1 = 1(x - 1) \Rightarrow y = x$. 13. **Solve integrations:** (a) $\int_0^1 e^x dx = [e^x]_0^1 = e - 1$. (b) $\int_0^1 x^3 dx = \left[ \frac{x^4}{4} \right]_0^1 = \frac{1}{4}$. (c) $\int (e^x + 3x^2) dx = e^x + x^3 + C$. 14. **Identify order and degree of** $\frac{d^2 y}{dx^2} = \frac{dy}{dx}$. Order is highest derivative order: 2. Degree is power of highest derivative (here 1). **Final answers:** 1. 4 2. $5 \ln(6)$ 3. 48 4. 7 5. 3 6. $12x^2$ 7. $7x^6 \sin x + x^7 \cos x$ 8. $e^x (5x^4 + x^5)$ 9. $e^{-x} (4x - 2x^2 - 3)$ 10. 2 11. $56x^6 + 30x^4 + 12x^2$ 12a. $y = x + 1$ 12b. $y = -x + 5$ 12c. $y = x$ 13a. $e - 1$ 13b. $\frac{1}{4}$ 13c. $e^x + x^3 + C$ 14. Order = 2, Degree = 1