This document is a mathematics exam or exercise sheet related to "Analyse 1" (Analysis 1), dated 31 October 2022, covering the following three exercises:

Exercise 1: Simplify the following expressions

1. $A = \arcsin(\sin(\frac{11\pi}{5}))$
2. $B = \arccos(\cos(\frac{11\pi}{4}))$
3. $C = \arctg(\tg(\frac{13\pi}{3}))$
4. $D = \cos(\arcsin(\frac{1}{2}))$
5. $E = \sin(\arccos(\frac{3}{5}))$

Exercise 2: Calculate, if they exist, the following limits

1. $\ell_1 = \lim_{x \to 0} \frac{|x|}{x}$
2. $\ell_2 = \lim_{x \to +\infty} \frac{\cos(x)}{x}$
3. $\ell_3 = \lim_{x \to +\infty} \left(1 + \frac{1}{x}\right)^x$
4. $\ell_4 = \lim_{x \to +\infty} \ln\left(1 + \frac{1}{x}\right) \sqrt{x}$
5. $\ell_5 = \lim_{x \to +\infty} \frac{\ln(1 + e^x)}{x}$
6. $\ell_6 = \lim_{x \to 0} \frac{\ln(1 + x) - x}{x^2}$

Exercise 3: Given the function $f$ defined by $f(x) = \arctg\left(\frac{1}{x^2}\right)$

1. Find the domain of definition of $f$.
2. Show that $f$ can be extended by continuity at 0. This extension is denoted as $g$.
3. Is this extension $g$ differentiable at 0?

Would you like a detailed solution for each exercise or a specific part? Here are some follow-up questions to consider:

1. How is the range of inverse trigonometric functions used in Exercise 1?
2. Why is the limit $\ell_3$ known as Euler's number?
3. What is the significance of logarithmic limits like $\ell_4$ and $\ell_5$?
4. How do you determine the continuity of a function at a point?
5. Why is the absolute value function important in the limit $\ell_1$?

Tip: For simplifying expressions involving inverse trigonometric functions, always consider the principal values of the functions.