The document contains two exercises in mathematics. I will provide a solution framework for each exercise and their subquestions based on the uploaded image.

Exercice 1

1. True or False Justifications:
   • (1) The function $x \mapsto \frac{|x|}{\sqrt{x^2+1}}$ is bounded on $\mathbb{R}$.
     • Analyze the behavior of $\frac{|x|}{\sqrt{x^2+1}}$ as $x \to \pm \infty$ and $x = 0$.
   • (2) If $f^2$ is increasing on $\mathbb{R}$, then $f$ is increasing on $\mathbb{R}$.
     • Counterexamples (e.g., $f(x) = -x$) can show whether this is true.
   • (3) $\lim_{x \to \infty} \frac{1+x}{x-1} = -1$.
     • Evaluate using algebraic simplification.
   • (4) Points $A$ and $B$ determine if $MAMB = 3MBMB$ forms a circle.
     • Use geometric reasoning.
   • (5) Set of points $M$ such that $\text{CM,AB} = \frac{\text{CD,AB}}{3} [2\pi]$.
     • This involves locus calculations.

Exercice 2

Part 1: Graph Analysis

1. Determine using the graph:
   • (a) Domain of $f(x)$: Find $x$ values for which $f(x)$ is defined and continuous.
   • (b) Asymptotes of $f(x)$: Analyze vertical and horizontal asymptotes.
   • (c) Limits $\lim_{x \to -\infty} f(x)$, $\lim_{x \to +\infty} f(x)$, and $f(-10^{22})$.
     • Read these values off the graph.
   • (d) Approximation of $f(-10^{22})$: Likely involves extrapolation.

Part 2: Compute Limits

1. Justify limits:
   • $\lim_{x \to +\infty} f(x)-2x+3$: Subtract dominant terms.
   • $\lim_{x \to +\infty} \frac{x-1}{f^2(x)-1}$: Factorize and simplify.
   • $\lim_{x \to +\infty} \frac{f(x)+3x}{x^{2021}+1}$: Analyze dominant power terms.
   • $\lim_{x \to 0^+} \frac{f(x)-2x}{f(x)+1}$: Use small $x$ approximation.

Part 3: Function $g(x) = \frac{x-1}{f(x)-1}$

1. Tasks:
   • (a) Domain of $g(x)$: Exclude points where $f(x) = 1$ or $x = 1$.
   • (b) Show $g$ is extendable at $x = 1$: Use the limit definition for continuity.
   • (c) Compute $\lim_{x \to +\infty} g(x)$: Analyze the fraction behavior.
   • (d) Show $g(x)$ has a vertical asymptote: Determine for what $x$ values the denominator goes to zero.

Would you like me to solve each of these step by step or focus on specific questions? Let me know how I can assist further! Here are 5 related questions to extend understanding:

1. What are the implications of a function being bounded on $\mathbb{R}$?
2. Can you find examples where $f^2(x)$ is increasing but $f(x)$ is not?
3. How do horizontal asymptotes reflect the behavior of functions as $x \to \pm \infty$?
4. What strategies can be used to simplify complex rational limits?
5. How do we rigorously determine the continuity of a function at a point?

Tip: Always verify your results graphically where possible to confirm the behavior of functions.