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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.

Explore solutions to mathematical problems involving limits, derivatives, integrals, and differential equations. Detailed step-by-step solutions provided for each problem. Suitable for undergraduate students and anyone interested in calculus and differential equations.

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