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The document you uploaded contains three exercises on calculus and algebra. Here’s an overview of each exercise:

### Exercise 1 (12 Points)

1. **Limits**:
   - (a) \( \lim_{x \to 3} \frac{x^2 + 2x + 10 - 4\sqrt{x^4 + 6x - 1}} \)
   - (b) \( \lim_{x \to +\infty} \frac{x^2 + x + \sqrt{x^2 - 3} - 2x}{x} \)
   - (c) \( \lim_{x \to \infty} \frac{\sqrt{x} - 3}{\sqrt{x} - 3} \)
   - (d) \( \lim_{x \to 1} \frac{\sqrt[3]{3x + 24} - 3}{x - 1} \)
   - (e) \( \lim_{x \to 2} \frac{x^2 - 4}{x - 2} \)
   - (f) \( \lim_{x \to 1} \frac{\sqrt{x - 1} - \sqrt{x - 1}}{x} \)

2. **Simplification**:
   - (a) \( \frac{\sqrt[3]{27} \times \sqrt{9} \times \sqrt{3}}{\sqrt{81} \times \sqrt{9}} \)
   - (b) \( \frac{\sqrt[5]{8} \times \sqrt[4]{16} \times \sqrt[6]{64}}{\sqrt[3]{2} \times \sqrt[2]{4}} \)

3. **Solve Equations**:
   - (a) Solve for \( x \) in \( \sqrt{1 - x + x + 1} = \sqrt{2} \).
   - (b) Solve for \( x \) in \( \sqrt{(x + 3)^2} + \sqrt{(3 - x)^2} = 2\sqrt{9 - x^2} \).

4. **Continuity**:
   - Prove that the function \( f(x) = \frac{\sqrt{x^2 + 2x + 3} + \frac{x - 1}{x^2 + 2x + 3}} \) is continuous over \( \mathbb{R} \).

### Exercise 2 (2 Points)

Given a piecewise function \( f(x) \) defined as:
   \[
   f(x) = \begin{cases} 
      \frac{x^2 - 5x + 6}{\sqrt{x + 1} - 2} & x > 3 \\
      \frac{x^2 - 3x - 18}{x^2 - 9} & x < 3 \\
      4 & x = 3 
   \end{cases}
   \]
1. Determine \( D_f \) and study the continuity of \( f \) at \( x = 3 \).

### Exercise 3 (6 Points)

Given a function \( g \) defined over the interval \( I = [0, +\infty) \) as:
   \[
   g(x) = x + \sqrt{x - 4}
   \]
1. Show that \( g \) is continuous over \( I \).
2. Study the variation (monotonicity) of \( g \) over \( I \).
3. Show that the equation \( g(x) = 0 \) has a unique solution \( \alpha \) within \( ]2, 3] \).
4. Verify that \( \alpha^2 - 9\alpha + 16 = 0 \), and determine the exact value of \( \alpha \).
5. Show that \( g \) has an inverse function \( g^{-1} \) defined over a subset \( J \subset I \).
6. Determine \( g^{-1}(x) \) over \( J \).

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### Related Questions:
1. How do you approach calculating limits involving radicals?
2. What are common techniques to prove continuity over an interval?
3. How can piecewise functions be analyzed for continuity?
4. What are the steps to determine if a function has an inverse?
5. How does one solve an equation involving nested square roots?

### Tip:
When solving limits with radicals, try rationalizing the expression to simplify. This can often help in evaluating limits at points where direct substitution leads to indeterminate forms.

Solve the calculus and algebra exercises provided in the image. The exercises involve calculating limits, simplifying expressions, solving equations, and proving continuity.

Explore challenging calculus and algebra exercises, including calculating limits, simplifying expressions, solving equations, and proving the continuity of functions. This guide covers techniques like rationalization and properties of continuity for piecewise functions, suitable for Grade 12 students.

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