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The image contains four mathematical problems asking to demonstrate the possibility of extending each given function \( f(x) \) continuously at specific points \( x_0 \). Here are the details from the image:

**Exercise 5.0: Prolongement par Continuité (Extension by Continuity)**

For each of the following cases, show that the function \( f \) admits an extension by continuity at \( x_0 \), then provide this extension.

1. \( f(x) = \frac{x\sqrt{x-1}}{\sqrt{3x+1} - \sqrt{x+3}} \quad \text{and} \quad x_0 = 1 \)

2. \( f(x) = \frac{\sqrt{x+6} + \sqrt{2x+5} - 3}{4 - x^2} \quad \text{and} \quad x_0 = -2 \)

3. \( f(x) = \frac{\tan(x) - \sin(x)}{x + \sin(x)} \quad \text{and} \quad x_0 = 0 \)

4. \( f(x) = \frac{\cos(x) - \sqrt{1 + \sin(x)}}{x} \quad \text{and} \quad x_0 = 0 \)

### Steps to solve:
For each function:
- Check for potential indeterminate forms (e.g., \( \frac{0}{0} \)) at the given point \( x_0 \).
- Apply limits as \( x \to x_0 \).
- If the limit exists, this provides the value of the continuous extension at that point.

Would you like to go through the solutions for these one by one?

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Here are 5 relative questions to deepen your understanding:
1. What is the importance of continuous extensions in real analysis?
2. How do limits help in determining the continuity of a function at a point?
3. What are common techniques for resolving indeterminate forms?
4. How do radical and trigonometric functions affect continuity?
5. How would you approach extending functions in complex analysis?

**Tip**: Always check for indeterminate forms when extending a function by continuity, as resolving these forms often helps find the continuous extension.

For each of the following cases, show that the function f admits an extension by continuity at x_0, then provide this extension: 1) f(x) = (x√(x−1)) / (√(3x+1) − √(x+3)) with x_0 = 1 2) f(x) = (√(x+6) + √(2x+5) − 3) / (4 − x^2) with x_0 = −2 3) f(x) = (tan(x) − sin(x)) / (x + sin(x)) with x_0 = 0 4) f(x) = (cos(x) − √(1 + sin(x))) / x with x_0 = 0

Explore how to extend functions by continuity at given points through limit calculations, handling indeterminate forms, and applying continuity and limit theorems. The examples involve trigonometric and radical functions, providing insight into advanced calculus techniques.

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