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The exercise asks to analyze the variation of four different functions based on their graphical representations (A, B, C, D). Each graph corresponds to a different function, and we are required to complete a variation table for each one, identifying key aspects such as monotonicity (increasing/decreasing), extrema, and limits as \( x \to \infty \), \( x \to -\infty \), and around vertical asymptotes if present.

### Analysis of the Graphs:

1. **Graph A**:
   - This resembles a cubic-like function with an inflection point.
   - **Behavior**: It decreases to a minimum, then increases again.
   - **Limits**: As \( x \to -\infty \), the function approaches \( -\infty \). As \( x \to \infty \), the function approaches \( \infty \).

2. **Graph B**:
   - This looks like a rational function with a vertical asymptote at \( x = 0 \).
   - **Behavior**: The function decreases toward the vertical asymptote from the left and increases after the asymptote.
   - **Limits**: As \( x \to -\infty \), the function approaches \( 0 \). As \( x \to \infty \), the function approaches \( \infty \). As \( x \to 0^- \), the function tends to \( -\infty \), and as \( x \to 0^+ \), the function tends to \( \infty \).

3. **Graph C**:
   - This is also a rational function with a vertical asymptote at \( x = 0 \), but it behaves differently from B.
   - **Behavior**: The function decreases on both sides of the asymptote.
   - **Limits**: As \( x \to -\infty \), the function tends to \( 0 \). As \( x \to \infty \), the function also tends to \( 0 \). As \( x \to 0^- \) and \( x \to 0^+ \), the function tends to \( -\infty \).

4. **Graph D**:
   - Another rational function with vertical asymptotes, but with more complex behavior.
   - **Behavior**: The function has a vertical asymptote at \( x = 0 \), increasing before and after the asymptote.
   - **Limits**: As \( x \to -\infty \), the function tends to \( 0 \). As \( x \to \infty \), the function tends to \( 0 \). As \( x \to 0^- \), the function tends to \( -\infty \), and as \( x \to 0^+ \), it tends to \( \infty \).

To complete the "tableaux de variations," you would organize the intervals of monotonicity and limits for each function. Would you like further details or help filling in the variation tables for each graph?

Here are five related questions to help expand this topic:
1. What is the significance of vertical asymptotes in rational functions?
2. How do inflection points affect the variation of a function?
3. Can you explain the difference between local minima and global minima for functions?
4. How do limits at infinity help in understanding the behavior of a function?
5. What are the key characteristics of a cubic function and a rational function?

**Tip**: When analyzing functions, always look for critical points (where the derivative is zero or undefined) and asymptotic behavior to understand their variation.

On donne les courbes représentatives de 4 fonctions. Dresser le tableau de variations complet de chacune avec les limites.

In this problem, we analyze four graphical representations of functions to determine their variation tables. The task involves identifying the monotonicity, behavior at infinity, and limits near vertical asymptotes.

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