https://file.aiquickdraw.com/mathbot/file/172635725336306rc7ni9.jpg

Let's analyze the graph for the function \( g(x) \) and determine the behavior of the function as \( x \) approaches positive and negative infinity.

### Analysis of \( \lim_{x \to \infty} g(x) \)
From the graph, as \( x \) increases towards positive infinity, the function \( g(x) \) does not approach a single value but instead continues to oscillate without bound. The function appears to keep increasing and decreasing without settling to a particular value or repeating a pattern that might suggest a periodic function.

### Analysis of \( \lim_{x \to -\infty} g(x) \)
Similarly, as \( x \) approaches negative infinity, the graph of \( g(x) \) shows an oscillatory behavior with no clear limit. The function continues to increase and decrease without approaching any finite number or repeating any pattern.

### Conclusion
Based on the visual observation:
- \( \lim_{x \to \infty} g(x) \) does not exist, since the function continues to oscillate and does not approach any specific value or infinity in a predictable manner.
- \( \lim_{x \to -\infty} g(x) \) also does not exist for similar reasons.

Please let me know if you need further details or explanations. Here are some related questions you might consider:

1. How would the limits change if the function \( g(x) \) had horizontal asymptotes?
2. What conditions would make \( \lim_{x \to \infty} g(x) \) exist for an oscillating function?
3. Can periodic functions have limits as \( x \) approaches infinity or negative infinity?
4. What are some real-world phenomena that could be modeled by a function similar to \( g(x) \)?
5. How does the behavior of \( g(x) \) at infinity impact the overall properties of the function?

**Tip:** When analyzing limits graphically, look for patterns such as approaching a particular value, infinity, or oscillation, which can provide clues about the limit's behavior.

For the function g whose graph is given, state the following: (a) lim x → ∞ (g(x)) and (b) lim x → -∞ (g(x)).

Analyze the behavior of the function g(x) graphically as x approaches positive and negative infinity. This problem covers concepts of limits, oscillations, and graphical interpretation. Suitable for students in Grades 11-12.

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