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This document appears to be a math assignment with exercises focused on limits, continuity, differentiability, and the behavior of functions. It is aimed at advanced high school students studying calculus, covering detailed analysis of functions and their derivatives. Here’s a breakdown of each exercise and sub-question:

### Exercise 1
Calculating limits as \( x \) approaches certain values:
1. Three different limit problems involving cube roots, square roots, and polynomial expressions.

### Exercise 2
Given a piecewise function \( f \), with different expressions for \( x \geq 0 \) and \( x < 0 \):
1. **(a) - (c)**: Limits as \( x \to 0 \), checking continuity at 0, and extending continuity on \( \mathbb{R} \).
2. **Differentiability**:
   - Study the differentiability of \( f \) from both sides of 0 and provide geometrical interpretations.
3. **Calculations**:
   - Calculating \( f'(x) \) on specific intervals and analyzing function variations.
4. **Inverse Functions**:
   - Show that a restriction of \( f \) has an inverse and calculate this inverse for certain values.

### Exercise 3
Working with another function defined on \( [0, +\infty) \):
1. **Limit Calculation** as \( x \to +\infty \).
2. **Derivative Analysis**:
   - Show that the derivative is negative, proving that \( f \) is strictly decreasing.
3. **Unique Solution of Equations**:
   - Demonstrate the existence and uniqueness of solutions for equations involving \( f(x) \) within specified intervals.
4. **Reciprocal Verification** and additional calculations on differentiability of the inverse.

Would you like assistance with specific sub-questions or detailed solutions for each exercise? 

### Follow-up Questions:
1. Would you like an in-depth explanation of limits and how they are solved?
2. Do you want guidance on continuity proofs and examples?
3. Are you interested in learning about geometric interpretations of differentiability?
4. Would you like to explore the concept of inverse functions and how to find them?
5. Do you need help with solving equations within given intervals?

### Tip:
When approaching these exercises, start by analyzing each function's behavior at critical points (e.g., points where the function's definition changes) to simplify the calculations and interpretations.

Advanced calculus exercise sheet focusing on limits, continuity, differentiability, and inverse functions for high school students.

Explore advanced calculus exercises on limits, continuity, and differentiability. This high school assignment covers detailed analysis of functions, including geometric interpretations and inverse functions, perfect for Grade 12 students preparing for calculus exams.

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