lim as x approaches infinity of: ln(3x)/9x

To evaluate the limit

\[
\lim_{{x \to \infty}} \frac{\ln(3x)}{9x},
\]

we'll analyze the behavior of the numerator and denominator as \( x \to \infty \).

### Step-by-Step Solution

1. **Analyze Growth Rates**: 
   - The numerator, \( \ln(3x) \), grows very slowly as \( x \to \infty \), because logarithmic functions grow slower than polynomial or exponential functions.
   - The denominator, \( 9x \), grows linearly and becomes infinitely large as \( x \to \infty \).

2. **Apply L'Hôpital's Rule**:
   Since this limit is of the form \( \frac{\infty}{\infty} \), we can apply L'Hôpital's Rule, which states that if the limit results in an indeterminate form \( \frac{\infty}{\infty} \), we can differentiate the numerator and the denominator and then take the limit of the resulting expression.

   Differentiate the numerator and denominator with respect to \( x \):
   - The derivative of \( \ln(3x) \) is \( \frac{1}{3x} \cdot 3 = \frac{1}{x} \).
   - The derivative of \( 9x \) is \( 9 \).

   Therefore, the limit becomes:
   \[
   \lim_{{x \to \infty}} \frac{\frac{1}{x}}{9} = \lim_{{x \to \infty}} \frac{1}{9x}.
   \]

3. **Evaluate the New Limit**:
   Now, as \( x \to \infty \), \( \frac{1}{9x} \to 0 \) because \( 9x \) grows infinitely large.

   Thus,
   \[
   \lim_{{x \to \infty}} \frac{\ln(3x)}{9x} = 0.
   \]

### Conclusion
\[
\lim_{{x \to \infty}} \frac{\ln(3x)}{9x} = 0.
\]

Would you like more details or have any questions?

---

1. What are typical applications of limits in calculus?
2. How does L'Hôpital's Rule work for other indeterminate forms like \( 0/0 \)?
3. Why do logarithmic functions grow slower than linear functions?
4. Can L'Hôpital's Rule be used more than once on a limit?
5. What other methods could be used if L'Hôpital's Rule were not applicable?

**Tip**: In limits involving infinity, compare growth rates (polynomial, exponential, logarithmic) to see if terms dominate or vanish.

Explore the limit of ln(3x)/9x as x approaches infinity with a step-by-step solution using L'Hôpital's Rule. This solution includes analysis of growth rates for logarithmic and linear functions and applies key calculus concepts, suitable for high school and college students.

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