The image contains mathematical problems, primarily covering calculus topics like limits, sequences, series, derivatives, continuity, Taylor expansions, and definite integrals. Let me know which specific question you’d like me to solve or explain in detail, or if you want a full walkthrough of the problems!

Here’s a breakdown of the main questions on the image:

1. Question 1: Limits and Sequences

   • (a) Find specific terms of a sequence $(x_n)$, given its recurrence relation, and calculate $\lim_{n \to \infty} x_n$.
   • (b) Evaluate a given limit involving trigonometric and algebraic expressions.
   • (c) Determine the value of $m$ to make a piecewise function continuous at $x = 0$.

2. Question 2: Derivatives and Taylor Expansions

   • (a) Compute a high-order derivative ($f^{(2024)}(0)$) for a given function.
   • (b) Expand a logarithmic function using Taylor series.
   • (c) Prove a given inequality involving logarithms.

3. Question 3: Definite Integrals

   • Solve two integrals:
     • (a) $\int_{0}^{\ln 3} \frac{e^x}{(e^x + 1)^3} dx$
     • (b) $\int \frac{x}{x^6 + 1} dx$

4. Question 4: Rolle’s Theorem and Continuity

   • (a) Determine $m$ for a quadratic function to satisfy Rolle’s theorem and find the critical point $c$.
   • (b) Check the continuity of a piecewise function involving $e^x$.

Let me know where you want to begin! Would you like a solution for a specific part? Or a detailed explanation for all?

Follow-up ideas:

1. Do you need help solving recurrence relations for sequences?
2. Should we explore Taylor series expansions and derivatives in detail?
3. Do you need a refresher on proving inequalities using calculus?
4. Would you like to analyze the integral computation steps?
5. Need assistance with Rolle's theorem and continuity conditions?

Tip: For problems involving piecewise continuity, always evaluate one-sided limits and ensure they match the function's value at the point!