This is a math exam sheet involving analysis of functions, sequences, and their properties. Let’s go step by step and summarize the questions for clarity:

Exercise 1 (12.5 points)

Part I: Function Analysis

• A function $f(x) = \frac{x+1}{\sqrt{x}}$ is defined on $[0, +\infty[$. Questions include:
  1. Limit calculations:
     • $\lim_{x \to +\infty} f(x)$ and interpretation of the result.
     • $\lim_{x \to 0^+} f(x)$, $\lim_{x \to 0^+} \frac{f(x)}{x}$, and their interpretations.
  2. Proofs and curve analysis:
     • Prove $f(x) - x = \frac{(1-\sqrt{x})(2x+\sqrt{x}+1)}{2\sqrt{x}}$.
     • Determine the position of $(C_f)$ (curve) relative to the line $y = x$.
  3. Monotonicity and derivative:
     • Prove $f'(x) = \frac{x-1}{2x^{3/2}}$ and analyze its behavior.
     • Create a variation table for $f(x)$.
  4. Concavity:
     • Prove $f''(x) = \frac{3-x}{4x^{5/2}}$.
     • Study concavity, points of inflection, and sketch $(C_f)$ with respect to the axes.

Part II: Sequence Analysis

• A sequence $(u_n)$ is defined as $u_0 = 4$, $u_{n+1} = f(u_n)$.
  1. Prove $u_n \geq 1 \, \forall n$.
  2. Show $(u_n)$ is decreasing.
  3. Prove $(u_n)$ converges and find its limit.

Exercise 2 (7.5 points)

Sequence Analysis

• A sequence $(U_n)$ is defined by $U_0 = 3$, $U_{n+1} = \frac{8(U_n - 1)}{U_n + 2}$.
  1. Prove $\forall n \in \mathbb{N}, 2 < U_n < 4$.
  2. Show $(U_n)$ is increasing, and deduce convergence.
  3. Analyze another sequence $(V_n)$:
     • Prove it is geometric, specifying the ratio $q$ and first term.
     • Derive its general expression.
     • Calculate $\lim_{n \to \infty} V_n$.
  4. Show $\forall n \geq 3, |U_{n+1} - 4| \leq \frac{4}{5} |U_n - 4|$.
  5. Prove $|U_n - 4| \leq \left(\frac{4}{5}\right)^n$.
  6. Deduce $\lim_{n \to \infty} U_n$.

Would you like me to:

• Solve any specific question?
• Provide hints or detailed explanations?

Related Questions:

1. What is the significance of a function's limits at 0 and infinity in analyzing its behavior?
2. How do you determine whether a sequence converges or diverges?
3. What is the role of the derivative in identifying monotonicity of functions?
4. How do you interpret a geometric sequence in the context of convergence?
5. Why is concavity important when sketching the graph of a function?

Tip: Start by solving the limit problems, as they often lay the foundation for deeper analyses in such exercises.