The image contains mathematical exercises in Arabic, focusing on limits, continuity, and function analysis. Here’s a breakdown of each exercise:

التمرين الأول (Exercise 1)

1. Calculate the following limits: $\lim_{x \to 2} \frac{x^3 - 8}{2x^2 - 4}$ $\lim_{x \to \infty} \frac{x^2 + 2x + 2}{3x - \sqrt{x + 1}}$

التمرين الثاني (Exercise 2)

1. Define the piecewise function: 2x + 3 & \text{if } x < 1 \\ x + a & \text{if } x \geq 1 \end{cases}$$ Find the value of $$a$$ that makes $$f$$ continuous over the interval $$[1, +\infty[$$.
2. Calculate the limit: $\lim_{x \to 1^+} f(x)$
3. Determine if $f$ is bounded on $\mathbb{R}$.

التمرين الثالث (Exercise 3)

1. Let: $g(x) = \frac{\sqrt[3]{x} - \sqrt{2}}{x-1}$ Calculate $g(1)$.
2. Show that the domain of $g$ is $D = [0, 2]$, and examine the function over the intervals $]1, 2]$ and $[0, 1[$.
3. Find the limits: $\lim_{x \to 1^-} g(x) \quad \text{and} \quad \lim_{x \to 1^+} g(x)$

التمرين الرابع (Exercise 4)

1. Define the function: $F(x) = \frac{x}{2\sqrt{x-1}}$
   • Find the domain of $F$ and calculate: $\lim_{x \to 1^+} F(x)$
2. Show that: $\lim_{x \to +\infty} \frac{F(x)}{x} = +\infty$
3. Differentiate $F$ over the interval $]1, +\infty[$.
4. Solve for: $h(x) = F(x), \quad h' \text{ in terms of its derivative over the domain of } h$

Would you like detailed solutions for any of these exercises, or do you have specific questions about them?

Related Questions:

1. How do you find the continuity of a piecewise function at a given point?
2. What techniques can be used to solve limits involving indeterminate forms?
3. How can you determine the boundedness of a function over an interval?
4. What are the methods for finding the derivative of a function involving radicals?
5. How do you analyze the behavior of functions at discontinuities?

Tip: When solving limits analytically, consider factoring, conjugates, or L'Hôpital's rule if you encounter indeterminate forms.