Let's go through the mathematical problems presented in the image. I'll provide solutions and explanations for each question.

Question 1: $\lim_{x \to 2} \frac{x^2-4}{x-2}$

Solution: $\lim_{x \to 2} \frac{x^2-4}{x-2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2}$ $= \lim_{x \to 2} (x+2) = 2 + 2 = 4$

Question 2: Find the derivative of $f(x) = 3x^3 + 5x - 2$

Solution: The derivative $f'(x)$ of the function $f(x) = 3x^3 + 5x - 2$ is calculated as follows: $f'(x) = \frac{d}{dx}(3x^3 + 5x - 2) = 3 \cdot 3x^2 + 5 \cdot 1 = 9x^2 + 5$

Question 3: Solve the differential equation $\frac{dy}{dx} = e^{x} \sin(x)$

Solution: To solve $\frac{dy}{dx} = e^{x} \sin(x)$, we need to integrate both sides with respect to $x$: $y = \int e^x \sin(x) \, dx$ Using integration by parts or recognizing it as a standard integral: $y = \frac{1}{2} e^x (\sin(x) - \cos(x)) + C$ where $C$ is the constant of integration.

Question 4: Compute the definite integral $\int_{0}^{\pi/2} \sin^2(x) \, dx$

Solution: To compute $\int_{0}^{\pi/2} \sin^2(x) \, dx$, we use the trigonometric identity $\sin^2(x) = \frac{1 - \cos(2x)}{2}$: $\int_{0}^{\pi/2} \sin^2(x) \, dx = \int_{0}^{\pi/2} \frac{1 - \cos(2x)}{2} \, dx$ $= \frac{1}{2} \int_{0}^{\pi/2} 1 \, dx - \frac{1}{2} \int_{0}^{\pi/2} \cos(2x) \, dx$ [ = \frac{1}{2} \left[ x \right]{0}^{\pi/2} - \frac{1}{2} \left[ \frac{\sin(2x)}{2} \right]{0}^{\pi/2} ] $= \frac{1}{2} \left( \frac{\pi}{2} - 0 \right) - \frac{1}{2} \left( 0 - 0 \right)$ $= \frac{\pi}{4}$

Do you have any questions or would you like further details on any of these solutions?

Here are five more questions you might find useful:

1. How do you find the integral of $e^{2x}$?
2. What is the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0?
3. How do you solve the differential equation $\frac{dy}{dx} + y = x$?
4. What is the Taylor series expansion of $e^x$ around $x = 0$?
5. How do you find the eigenvalues of a 2x2 matrix?

Tip: When solving differential equations, always check if they can be simplified using substitution or standard forms before applying more complex methods.