https://file.aiquickdraw.com/mathbot/file/1725843673813lp2rrx9y.jpg

The image contains a graph of a function \( f(x) \) and a set of questions requesting specific limits. Let me analyze the content of the graph and questions to provide the limits.

From the graph, I can identify that the function exhibits oscillatory behavior on the left and has an undefined behavior near \( x = 0 \), potentially indicating a vertical asymptote.

Here are the questions from the image:

1. \( \lim_{x \to 0^-} f(x) \)
2. \( \lim_{x \to 0^+} f(x) \)
3. \( \lim_{x \to \infty} f(x) \)
4. \( \lim_{x \to -\infty} f(x) \)

### Analysis:
1. **\( \lim_{x \to 0^-} f(x) \)**: As \( x \) approaches 0 from the left, the function seems to decrease sharply, likely tending towards \( -\infty \). Thus, \( \lim_{x \to 0^-} f(x) = -\infty \).
   
2. **\( \lim_{x \to 0^+} f(x) \)**: As \( x \) approaches 0 from the right, the function tends upwards, heading towards \( +\infty \). So, \( \lim_{x \to 0^+} f(x) = \infty \).

3. **\( \lim_{x \to \infty} f(x) \)**: As \( x \) tends to infinity, the function appears to oscillate but overall approaches a horizontal asymptote, likely at \( y = 0 \). Therefore, \( \lim_{x \to \infty} f(x) = 0 \).

4. **\( \lim_{x \to -\infty} f(x) \)**: As \( x \) tends to negative infinity, the function shows oscillatory behavior but seems to stabilize and approach 0. Hence, \( \lim_{x \to -\infty} f(x) = 0 \).

### Final Answers:
1. \( \lim_{x \to 0^-} f(x) = -\infty \)
2. \( \lim_{x \to 0^+} f(x) = \infty \)
3. \( \lim_{x \to \infty} f(x) = 0 \)
4. \( \lim_{x \to -\infty} f(x) = 0 \)

Would you like further details on any of these limits or their calculations?

### Relative questions:
1. How is a vertical asymptote defined in a function's behavior?
2. What is the difference between left-hand and right-hand limits?
3. Can a function have oscillatory behavior but still have a defined limit at infinity?
4. Why do some functions have infinite limits at specific points?
5. What does it mean when a function approaches zero as \( x \to \infty \)?

### Tip:
When analyzing limits at infinity, always consider both horizontal asymptotes and any oscillatory behavior, as these factors can affect the limit.

What are the following limits based on the graph of the function provided: 1. lim(x → 0^-) f(x), 2. lim(x → 0^+) f(x), 3. lim(x → ∞) f(x), 4. lim(x → -∞) f(x)?

Analyze the graph of a function to determine the left-hand and right-hand limits at x = 0, as well as the limits as x approaches infinity and negative infinity. This problem covers key calculus concepts, including asymptotic behavior and oscillations.

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