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?

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.