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The problem states that \( h \) is a function defined on the interval \( ]0, +\infty[ \) and satisfies the inequality:

\[
x^3 - 3x^2 < h(x) < x^3 + x^2 \quad \text{for all} \quad x \in ]0, +\infty[
\]

We are asked to calculate the following limits:

1. \( \lim_{x \to 0^+} h(x) \)
2. \( \lim_{x \to +\infty} h(x) \)
3. \( \lim_{x \to +\infty} \frac{h(x)}{x^3} \)

### 1. Limit as \( x \to 0^+ \) of \( h(x) \):

From the inequality:

\[
x^3 - 3x^2 < h(x) < x^3 + x^2
\]

As \( x \to 0^+ \), both bounds \( x^3 - 3x^2 \) and \( x^3 + x^2 \) approach \( 0 \):

\[
\lim_{x \to 0^+} (x^3 - 3x^2) = 0 \quad \text{and} \quad \lim_{x \to 0^+} (x^3 + x^2) = 0
\]

By the squeeze theorem:

\[
\lim_{x \to 0^+} h(x) = 0
\]

### 2. Limit as \( x \to +\infty \) of \( h(x) \):

For large \( x \), the bounds for \( h(x) \) behave as:

\[
x^3 - 3x^2 < h(x) < x^3 + x^2
\]

As \( x \to +\infty \), both bounds grow like \( x^3 \):

\[
\lim_{x \to +\infty} (x^3 - 3x^2) = +\infty \quad \text{and} \quad \lim_{x \to +\infty} (x^3 + x^2) = +\infty
\]

Thus, by the squeeze theorem:

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

### 3. Limit as \( x \to +\infty \) of \( \frac{h(x)}{x^3} \):

We divide the bounds for \( h(x) \) by \( x^3 \):

\[
\frac{x^3 - 3x^2}{x^3} < \frac{h(x)}{x^3} < \frac{x^3 + x^2}{x^3}
\]

Simplifying each term:

\[
1 - \frac{3}{x} < \frac{h(x)}{x^3} < 1 + \frac{1}{x}
\]

As \( x \to +\infty \), both bounds converge to 1:

\[
\lim_{x \to +\infty} \left( 1 - \frac{3}{x} \right) = 1 \quad \text{and} \quad \lim_{x \to +\infty} \left( 1 + \frac{1}{x} \right) = 1
\]

By the squeeze theorem:

\[
\lim_{x \to +\infty} \frac{h(x)}{x^3} = 1
\]

### Final Results:

1. \( \lim_{x \to 0^+} h(x) = 0 \)
2. \( \lim_{x \to +\infty} h(x) = +\infty \)
3. \( \lim_{x \to +\infty} \frac{h(x)}{x^3} = 1 \)

Would you like any additional details or clarifications?

Here are 5 related questions you could explore:
1. How does the squeeze theorem work in limit problems?
2. What would happen if the bounds on \( h(x) \) were different?
3. Can \( h(x) \) have any discontinuities within the interval \( ]0, +\infty[ \)?
4. How do asymptotic behaviors affect limit calculations?
5. What is the behavior of \( h(x) \) at specific points within the interval?

**Tip:** When using the squeeze theorem, make sure that the upper and lower bounds approach the same limit.

h est une fonction numérique définie sur l’intervalle ]0 ; +∞[ et vérifie : ∀x ∈ ]0 ; +∞[, x³ - 3x² < h(x) < x³ + x². Calculer les limites suivantes : lim_{x -> 0^+} h(x), lim_{x -> +∞} h(x) et lim_{x -> +∞} [h(x) / x³] (justifier vos réponses).

Calculate the limits of h(x) defined within the interval ]0, +∞[ and bounded by x³ - 3x² and x³ + x². This problem involves the Squeeze Theorem to determine the limits as x approaches 0 and infinity. Suitable for advanced high school and undergraduate students.

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