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The image shows a question involving limits and a graph with vertical asymptotes at \( x = -3 \) and \( x = 4 \). The task is to analyze the limits at these points. Let's walk through the limits based on the graph provided:

1. **Limit as \( x \to -3 \)**:
   The graph shows a vertical asymptote at \( x = -3 \). For vertical asymptotes, the limit does not exist (DNE), as the function tends towards infinity or negative infinity. Depending on which direction you approach from (left or right), the limit might tend toward \( +\infty \) or \( -\infty \).

   - \( \lim_{x \to -3^-} f(x) \) (approaching from the left): The function appears to go to \( -\infty \).
   - \( \lim_{x \to -3^+} f(x) \) (approaching from the right): The function appears to go to \( +\infty \).
   So the limit at \( x = -3 \) does not exist because it tends to different values from either side.

2. **Limit as \( x \to 4 \)**:
   Similarly, there's a vertical asymptote at \( x = 4 \). The limit does not exist because the function likely diverges to \( \infty \) or \( -\infty \).

   - \( \lim_{x \to 4^-} f(x) \): The function seems to go to \( +\infty \).
   - \( \lim_{x \to 4^+} f(x) \): The function appears to go to \( -\infty \).
   So again, the limit at \( x = 4 \) does not exist for similar reasons as \( x = -3 \).

3. **Other Limits**:
   Based on the question setup, you may also need to analyze left and right limits explicitly, and the general behavior around these points (which involves checking for one-sided limits).

Would you like a more detailed explanation of any specific limit or behavior? 

Here are 5 related questions:
1. What are one-sided limits, and how do they affect whether a limit exists?
2. How can vertical asymptotes influence the behavior of limits?
3. Can a limit exist if the function approaches infinity on both sides of a point?
4. How do we handle limits at points where the function is discontinuous?
5. What is the relationship between limits and horizontal asymptotes?

**Tip**: Always check the behavior of the function from both sides (left and right) of a point when determining the existence of a limit!

The graph of f in the figure has vertical asymptotes at x = -3 and x = 4. Analyze the following limits.

This problem involves determining the limits of a function at vertical asymptotes (x = -3 and x = 4). Learn how to approach one-sided limits and analyze function behavior at these critical points. Suitable for students studying high school calculus or pre-calculus.

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