The natural logarithm function, denoted by \( \ln(x) \), has several important limits that are useful in calculus. Here are some key limits involving the natural logarithm:

1. **Limit as \( x \) approaches 0 from the right:**
   \[
   \lim_{x \to 0^+} \ln(x) = -\infty
   \]
   This means that as \( x \) gets closer to 0 from the positive side, the value of \( \ln(x) \) decreases without bound.

2. **Limit as \( x \) approaches infinity:**
   \[
   \lim_{x \to \infty} \ln(x) = \infty
   \]
   This indicates that as \( x \) increases indefinitely, \( \ln(x) \) also increases without bound.

3. **Limit as \( x \) approaches 1:**
   \[
   \lim_{x \to 1} \ln(x) = 0
   \]
   At \( x = 1 \), the natural logarithm equals zero.

4. **Limit as \( x \) approaches infinity for \( \frac{\ln(x)}{x} \):**
   \[
   \lim_{x \to \infty} \frac{\ln(x)}{x} = 0
   \]
   This shows that \( \ln(x) \) grows more slowly than \( x \) itself as \( x \) becomes very large.

5. **Limit as \( x \) approaches infinity for \( \frac{x}{\ln(x)} \):**
   \[
   \lim_{x \to \infty} \frac{x}{\ln(x)} = \infty
   \]
   This indicates that \( x \) grows much faster than \( \ln(x) \) as \( x \) increases.

6. **Limit as \( x \) approaches 0 from the right for \( x\ln(x) \):**
   \[
   \lim_{x \to 0^+} x\ln(x) = 0
   \]
   This is a result of \( x \) approaching 0 faster than \( \ln(x) \) tends to \( -\infty \).

These limits are fundamental in various applications in calculus, especially in evaluating the behavior of logarithmic functions near important points.

Would you like further details on any of these limits or have any questions?

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**Here are 8 related questions:**

1. What is the derivative of the natural logarithm function?
2. How does the limit \( \lim_{x \to \infty} \ln(x^2) \) compare to \( \lim_{x \to \infty} \ln(x) \)?
3. What are the limits involving the logarithm function when approaching negative infinity?
4. How can L'Hôpital's rule be applied to evaluate limits involving \( \ln(x) \)?
5. What is the limit \( \lim_{x \to 0^+} \ln(x^2) \)?
6. Can you explain the relationship between \( \ln(x) \) and the exponential function \( e^x \)?
7. What is the limit of \( \ln(x+1) - \ln(x) \) as \( x \) approaches infinity?
8. How does the limit of \( \ln(x) \) as \( x \) approaches 0 from the left differ from approaching 0 from the right?

**Tip:** When dealing with limits involving logarithmic and exponential functions, remember that the logarithm grows very slowly compared to polynomials and exponential functions.

Explore the limits of the natural logarithm function (ln), including limits as x approaches 0, infinity, and 1. Learn about the behavior of ln(x) and its applications in calculus.

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