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.