This is an analysis (calculus) exercise set with three main parts, covering topics such as limits, trigonometric identities, and properties of functions, including continuity, differentiability, and domain analysis. Here’s a structured breakdown of each exercise:

Exercise 1: Limits

1. Calculate the limits:
   • $\ell_1 = \lim_{x \to +\infty} \frac{x^5 - 2x + 1}{x^6 + 2x + 4 + x}$
   • $\ell_2 = \lim_{x \to +\infty} \frac{\cos(x)}{x^3}$
   • $\ell_3 = \lim_{x \to 0} \frac{\operatorname{sh}(x)}{x}$
   • $\ell_4 = \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x$

Each of these limits requires different techniques, including asymptotic analysis, L'Hôpital's Rule, or recognizing standard limit forms.

Exercise 2: Trigonometric and Inverse Trigonometric Identities

1. Show that $\forall x \in [-1, 1]$: $\arccos(x) + \arcsin(x) = \frac{\pi}{2}$.
   • This result uses the relationship between the arccosine and arcsine functions.
2. Simplify expressions involving inverse trigonometric functions:
   • (a) $A = \arcsin\left(\sin\left(\frac{24\pi}{5}\right)\right)$
   • (b) $B = \arccos\left(\sin\left(\frac{24\pi}{5}\right)\right)$
   • (c) $C = \arccos\left(\cos\left(\frac{24\pi}{5}\right)\right)$
   • (d) $D = \arcsin\left(\cos\left(\frac{24\pi}{5}\right)\right)$

This part involves finding equivalent expressions for each function using periodicity and the ranges of inverse trigonometric functions.

Exercise 3: Function Analysis

1. Consider functions $f$ and $g$ defined by:

   • $f(x) = \arctan(x) + \arctan\left(\frac{1}{x}\right)$

   • $g(x) = \arcsin(x) f(x)$

   • (a) Determine the domain of definition for $f$.

   • (b) Determine the domain of differentiability for $f$ and calculate $f'$.

   • (c) Show that $f(x) = \begin{cases} \frac{\pi}{2} & \text{if } x > 0 \\ -\frac{\pi}{2} & \text{if } x < 0 \end{cases}$.

   • (d) Find the domain of definition for $g$.

   • (e) Show that $g$ can be extended continuously at $x = 0$. Denote this extension as $h$.

   • (f) Calculate $\lim_{x \to 0} \frac{\arcsin(x)}{x}$.

   • (g) Show that $h$ is not differentiable at $x = 0$.

This part involves understanding limits, continuity, differentiability, and the behavior of trigonometric functions around singularities.

Would you like a step-by-step solution for any of these exercises?

Here are related questions to help deepen understanding:

1. What are common methods for evaluating limits that approach infinity?
2. How can we use trigonometric identities to simplify expressions with inverse trigonometric functions?
3. How does continuity at a point differ from differentiability at that point?
4. What role does the range of inverse trigonometric functions play in simplifying expressions?
5. How can periodicity of trigonometric functions be applied to evaluate inverse trigonometric values?

Tip: When calculating limits, always check if you can simplify the expression before applying complex techniques like L'Hôpital's Rule.