I can see the mathematical problems from your image. I'll solve them step by step:

Problem c. $\lim_{x \to \infty} \left( \frac{3x - 1}{3x + 4} \right)^{2x}$

1. Simplify the fraction inside the limit: $\frac{3x - 1}{3x + 4} \approx 1 - \frac{5}{3x} \quad \text{as } x \to \infty.$

2. Rewrite the base as $e$-exponent: $\left(1 - \frac{5}{3x}\right)^{2x} = \exp\left(2x \ln\left(1 - \frac{5}{3x}\right)\right).$

3. Approximate $\ln(1 - u) \approx -u$ for small $u$: $\ln\left(1 - \frac{5}{3x}\right) \approx -\frac{5}{3x}.$

4. Substituting: $2x \ln\left(1 - \frac{5}{3x}\right) \approx 2x \cdot \left(-\frac{5}{3x}\right) = -\frac{10}{3}.$

5. Final answer: $\lim_{x \to \infty} \left( \frac{3x - 1}{3x + 4} \right)^{2x} = e^{-\frac{10}{3}}.$

Problem d. $\lim_{x \to -1} \frac{x^2}{1 - x^2}$

1. Substituting $x = -1$: $\text{The denominator becomes } 1 - (-1)^2 = 1 - 1 = 0.$

   This creates a $\frac{0}{0}$ indeterminate form.

2. Factorize and simplify around $x \to -1$: $\frac{x^2}{1 - x^2} = \frac{x^2}{(1 - x)(1 + x)}.$

3. Substituting $x = -1$ directly reveals a vertical asymptote, and the result diverges to $+\infty$ or $-\infty$ depending on the direction of approach: $\lim_{x \to -1} \frac{x^2}{1 - x^2} = +\infty \text{ or } -\infty.$

Problem h. $\lim_{x \to 0} \frac{\sin(3x)}{\sin(x)}$

1. Use the small-angle approximation: $\sin(3x) \approx 3x \quad \text{and} \quad \sin(x) \approx x \quad \text{as } x \to 0.$

2. Substitute: $\frac{\sin(3x)}{\sin(x)} \approx \frac{3x}{x} = 3.$

3. Final answer: $\lim_{x \to 0} \frac{\sin(3x)}{\sin(x)} = 3.$

Problem i. $\lim_{x \to 0} \frac{\tan(5x)}{\tan(x)}$

1. Use the small-angle approximation: $\tan(5x) \approx 5x \quad \text{and} \quad \tan(x) \approx x \quad \text{as } x \to 0.$

2. Substitute: $\frac{\tan(5x)}{\tan(x)} \approx \frac{5x}{x} = 5.$

3. Final answer: $\lim_{x \to 0} \frac{\tan(5x)}{\tan(x)} = 5.$

Would you like further explanations or a visual representation of any solution?

Here are 5 related questions to explore further:

1. How does the concept of limits work for exponential expressions like in c?
2. Can you explain the importance of small-angle approximations in trigonometric limits?
3. Why does the denominator in d create a vertical asymptote at $x = -1$?
4. How does L’Hôpital’s Rule apply to cases with $\frac{0}{0}$?
5. What are other cases where $\tan(x)$ or $\sin(x)$ approximations fail?

Tip: For exponential limits, rewriting the base using natural logarithms often simplifies the solution!