The exercise shown in the image is in Arabic, and it involves analyzing the function $f(x) = -x + \sqrt{x^2 + 8}$. Several parts of the exercise are outlined, involving limits, derivatives, and analysis of a curve. Here is a step-by-step breakdown based on what I see:

Problem Breakdown:

1. Limits of the function $f(x)$: The first task asks to find the limits of $f(x)$ at the edges of its domain, likely focusing on behavior as $x \to \infty$ and $x \to 0$.

2. Domain of the function: The second task is to determine the domain of $f(x)$. This will involve solving the inequality for $\sqrt{x^2 + 8} \geq 0$ and understanding where the square root is defined.

3. Derivative: The third part asks for the derivative of the function $f(x)$ and then to construct a table of variations, which is common in calculus to show intervals of increase and decrease.

4. Line Equation $y = -2x$: The fourth task mentions studying the relationship between the line $y = -2x$ and the curve $f(x)$. This is likely related to finding intersections or the slope of the tangent at specific points.

5. Asymptotic behavior: It seems part of the question asks for analyzing the asymptotic behavior of the curve.

Suggested Approach:

1. Limits:

   • Find $\lim_{x \to \infty} f(x)$ and $\lim_{x \to 0} f(x)$.

2. Domain:

   • Since the square root function is always defined for $x^2 + 8 \geq 0$, the domain of $f(x)$ would be $\mathbb{R}$ (all real numbers), as there are no restrictions for real values of $x$.

3. Derivative:

   • Use the chain rule to differentiate $f(x) = -x + \sqrt{x^2 + 8}$, then analyze where the derivative is positive or negative.

4. Intersections:

   • Solve for where $f(x) = -2x$ to find intersections between the curve and the line.

Let me know which part you would like to dive deeper into, or if you want help solving any specific part of the exercise!

Here are 5 related questions:

1. What is the behavior of $f(x)$ as $x \to -\infty$?
2. How do we find the intersection points between $f(x)$ and the line $y = -2x$?
3. How can we use the derivative of $f(x)$ to determine concavity?
4. What role do asymptotes play in analyzing the graph of $f(x)$?
5. How would you interpret the physical meaning of the function $f(x)$?

Tip: When differentiating functions involving square roots, always remember to apply the chain rule carefully!