This assignment includes three main exercises, each focusing on different areas of calculus and algebra.

Exercise 1: Limits, Simplification, and Continuity

1. Limits - Calculating several limits as variables approach specific values:

   • $a: \lim_{x \to 3} \frac{\sqrt{x^2 + 2x + 10} - 4}{x^2 + 6x - 1}$
   • $b: \lim_{x \to +\infty} \frac{\sqrt{x^2 + x + \sqrt{x^2 - 3 - 2x}}}{x}$
   • $c: \lim_{x \to +\infty} \frac{\sqrt{x} - 3}{\sqrt{x} - 3}$
   • $d: \lim_{x \to 1} \frac{\sqrt{3x + 24 - 3}}{x - 1}$
   • $e: \lim_{x \to 2} \frac{x^2 - 4}{\sqrt{x - 2}}$
   • $f: \lim_{x \to 1} \frac{\sqrt{x - 1} - \sqrt{x - 1}}{x}$

2. Simplification - Simplifying expressions involving roots and exponents:

   • $a: \frac{\sqrt[3]{27 \times 9 \times (\sqrt{3})^3}}{\sqrt{81 \times \sqrt{9}}}$
   • $b: \frac{\sqrt{8} \times \sqrt[6]{16} \times \sqrt[4]{64}}{\sqrt[2]{2} \times \sqrt[4]{4}}$

3. Equation Solving - Solving equations involving square roots and other algebraic expressions:

   • $a: \sqrt{1 - x + \sqrt{x + 1}} = \sqrt{2}$
   • $b: \sqrt{x + 3} + \sqrt{3 - x} = 2\sqrt{9 - x^2}$

4. Continuity of a Function - Proving continuity of the function: $f(x) = \frac{\sqrt{x^2 + 2x + 3} - \frac{x - 1}{x^2 + 2x + 3}}{x^2 + 2x + 3}$

Exercise 2: Piecewise Function Continuity

1. Function Definition and Continuity at $x = 3$: Given the piecewise function: \frac{x^2 - 5x + 6}{x + 1 - 2} & x > 3 \\ \frac{x^2 - 3x - 18}{x^2 - 9} & x < 3 \end{cases}$$ Determine the domain $$D_f$$ and study the continuity at $$x = 3$$.

Exercise 3: Properties of a Function

Given $g(x) = x + \sqrt{x - 4}$ over the interval $I = [0, +\infty)$:

1. Show that $g$ is Continuous on $I$.
2. Study the Variations of $g$ on $I$.
3. Prove that $g(x) = 0$ has a Unique Solution in the Interval $[2, 3]$.
4. Verify that $a^2 - 9a + 16 = 0$ and Find the Value of $a$.
5. Show that $g$ has an Inverse Function $g^{-1}$ on a Subinterval of $I$.
6. Determine $g^{-1}(x)$ for $x \in J$.

Would you like a detailed solution for a specific problem, or do you have any questions about any of the exercises?

Related Questions:

1. What techniques can be used to find limits involving square roots?
2. How do we determine the continuity of piecewise functions?
3. What are common methods for simplifying expressions with roots and powers?
4. How can we solve radical equations effectively?
5. What is the process for finding an inverse function of a given function?

Tip:

For piecewise functions, always verify that both parts meet at the boundary point to confirm continuity at that point.