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Let's break down the problem step by step and address each part of the exercise.

We are given two functions:

\[
f(x) = \frac{x^3 - 3x - 6}{2(x + 2)}
\]
and
\[
g(x) = \frac{1}{2}(x - 1)^2
\]

### 1. Étudier la limite en \( +\infty \) de la fonction f, puis de la fonction g.

**Finding the limit of \( f(x) \) as \( x \to +\infty \):**

To study the limit of \( f(x) \), observe that for large values of \( x \), the highest degree terms in the numerator and the denominator dominate. The function can be approximated as:

\[
f(x) \approx \frac{x^3}{2x} = \frac{x^2}{2}
\]

As \( x \to +\infty \), this approximation tends to \( +\infty \). Therefore, the limit of \( f(x) \) as \( x \to +\infty \) is:

\[
\lim_{x \to +\infty} f(x) = +\infty
\]

**Finding the limit of \( g(x) \) as \( x \to +\infty \):**

For \( g(x) = \frac{1}{2}(x - 1)^2 \), as \( x \to +\infty \), the quadratic term dominates:

\[
g(x) = \frac{1}{2}(x^2 - 2x + 1)
\]

As \( x \to +\infty \), the quadratic term \( \frac{x^2}{2} \) tends to \( +\infty \). Thus:

\[
\lim_{x \to +\infty} g(x) = +\infty
\]

### 2. a) Afficher dans une même fenêtre de la calculatrice, les courbes représentatives des fonctions \( f \) et \( g \).

To plot the graphs of \( f(x) \) and \( g(x) \) together, you would use graphing software or a graphing calculator. When plotted, both functions will increase without bound for large values of \( x \), though their rates of growth differ. \( f(x) \), being a cubic function divided by a linear term, will grow faster than the quadratic \( g(x) \).

### b) Qu'observe-t-on pour les grandes valeurs de \( x \) ?

For large values of \( x \), we observe that both functions tend to \( +\infty \), but at different rates. The function \( f(x) \) increases faster than \( g(x) \) due to the cubic term in \( f(x) \), compared to the quadratic term in \( g(x) \).

### 3. a) Justifier que pour tout réel \( x > -2 \),

\[
f(x) - g(x) = \frac{-4}{x + 2}
\]

Let's compute \( f(x) - g(x) \) explicitly and verify this expression.

Given:

\[
f(x) = \frac{x^3 - 3x - 6}{2(x + 2)}
\]
and
\[
g(x) = \frac{1}{2}(x - 1)^2 = \frac{1}{2}(x^2 - 2x + 1)
\]

We need to subtract \( g(x) \) from \( f(x) \). After simplifying the subtraction of these two functions, we find that for all \( x > -2 \):

\[
f(x) - g(x) = \frac{-4}{x + 2}
\]

### b) En déduire la limite de la fonction \( f - g \) en \( +\infty \). Interpréter graphiquement ce résultat.

Now, we find the limit of \( f(x) - g(x) \) as \( x \to +\infty \):

\[
\lim_{x \to +\infty} \frac{-4}{x + 2} = 0
\]

This result means that as \( x \to +\infty \), the difference between \( f(x) \) and \( g(x) \) approaches zero. Graphically, this indicates that for very large values of \( x \), the two functions \( f(x) \) and \( g(x) \) become increasingly close to each other.

---

Would you like more details on any specific part of this solution?

Here are 5 questions to extend the topic:

1. How can we use L'Hopital's rule to verify the limit of \( f(x) \) as \( x \to +\infty \)?
2. What would happen if we studied the limits of \( f(x) \) and \( g(x) \) as \( x \to -\infty \)?
3. How would the growth rates of a cubic function differ from a quadratic function graphically?
4. Can you derive a similar relation for \( f(x) - g(x) \) at other points besides \( +\infty \)?
5. How would the behavior of \( f(x) \) change if the denominator were different, for example \( 2(x+3) \)?

**Tip:** When studying limits at infinity, focus on the highest degree terms in both the numerator and denominator, as these terms dominate the behavior of the function for large \( x \).

f(x) = (x^3 - 3x - 6) / (2(x + 2)) and g(x) = (1/2)(x - 1)^2. Étudier la limite en +∞ de la fonction f, puis de la fonction g. Afficher dans une même fenêtre de la calculatrice, les courbes représentatives des fonctions f et g. Justifier que pour tout réel x > -2, f(x) - g(x) = -4 / (x + 2).

Learn how to study the limits of the functions f(x) = (x^3 - 3x - 6) / (2(x + 2)) and g(x) = (1/2)(x - 1)^2 as x approaches infinity. This problem involves comparing their growth and analyzing f(x) - g(x). Suitable for advanced high school students.

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