Draw the graph of sin 1/x and explain the steps

Here's the graph of \( f(x) = \sin\left(\frac{1}{x}\right) \). As expected, the function oscillates rapidly between -1 and 1 as \( x \) approaches 0 from both sides, and the oscillations become more gradual as \( |x| \) increases. The graph doesn't include \( x = 0 \) because the function is undefined there.

### Key Observations:
- For \( x \) close to 0, the graph oscillates infinitely, which is why it appears densely packed near the origin.
- For larger values of \( x \), the oscillations smooth out and resemble a standard sine curve but with decreasing frequency.

Would you like more details or have any questions? Here are some related questions to consider:

1. What happens if we try to evaluate the limit of \( \sin\left(\frac{1}{x}\right) \) as \( x \) approaches 0?
2. How does the behavior of \( f(x) = \sin\left(\frac{1}{x}\right) \) change when you introduce a shift or scaling factor?
3. Can you describe why the function is not continuous at \( x = 0 \)?
4. What other functions behave similarly near \( x = 0 \)?
5. How would the graph change if we used \( \cos\left(\frac{1}{x}\right) \) instead of \( \sin\left(\frac{1}{x}\right) \)?

**Tip**: Functions with oscillations near a point, like \( \sin\left(\frac{1}{x}\right) \), often have undefined limits at that point due to the rapid change in values.

Draw the graph of sin 1/x and explain the steps.

Explore the graph of the function f(x) = sin(1/x), understanding its rapid oscillations near zero and smoother behavior as x increases. Learn about its key features, including undefined behavior at x = 0, and how trigonometric concepts apply. This analysis is suitable for students in Grades 11-12 or early college.

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