https://file.aiquickdraw.com/mathbot/file/1726170386958brm9bmhr.jpg

is this negative infinity or positive infinity?

The limit in the image is:

\[
\lim_{{x \to 0^+}} \ln(\sin(x))
\]

To evaluate this limit, let’s break it down step by step:

1. **Behavior of \(\sin(x)\) as \(x \to 0^+\):**
   As \(x \to 0^+\), \(\sin(x)\) approaches 0 from the positive side. Specifically:
   \[
   \sin(x) \to 0^+ \text{ as } x \to 0^+.
   \]

2. **Behavior of \(\ln(x)\) as \(x \to 0^+\):**
   The natural logarithm function \(\ln(x)\) tends to \(-\infty\) as \(x \to 0^+\). That is:
   \[
   \ln(x) \to -\infty \text{ as } x \to 0^+.
   \]

3. **Combining the two:**
   Since \(\sin(x) \to 0^+\), and the logarithm of a number approaching 0 from the positive side goes to \(-\infty\), the overall limit becomes:
   \[
   \lim_{{x \to 0^+}} \ln(\sin(x)) = -\infty.
   \]

Therefore, the limit is **negative infinity**.

Would you like any more details or have any other questions?

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Here are 5 related questions to deepen understanding:
1. How does the limit behave if we evaluate \(\lim_{{x \to 0^-}} \ln(\sin(x))\)?
2. What happens if we change the function to \(\ln(\cos(x))\) as \(x \to 0^+\)?
3. Can you explain why \(\ln(x)\) tends to \(-\infty\) as \(x \to 0^+\)?
4. How does L'Hopital's Rule apply to this type of limit problem?
5. What is the limit of \(\sin(x)\) as \(x \to \infty\)?

**Tip**: When evaluating limits involving logarithms and trigonometric functions, always check how the inner function behaves as it approaches the point of interest!

Is this negative infinity or positive infinity?

Discover how to solve the limit lim_{x → 0^+} ln(sin(x)) and understand why the result is negative infinity. This advanced calculus problem explores the behavior of trigonometric and logarithmic functions as x approaches zero. Ideal for students in Grades 11-12.

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