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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.

This document is a mathematics exam or exercise sheet related to 'Analyse 1', covering exercises on simplifying expressions, calculating limits, and analyzing a function's continuity and differentiability.

Explore solutions to a first-year undergraduate 'Analyse 1' exam. Topics include simplifying inverse trigonometric expressions, evaluating complex limits, and analyzing a function's continuity and differentiability. Ideal for students in calculus or analysis.

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