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To solve this, let's complete the application of L'Hôpital's Rule to find the missing expression.

### Step-by-Step Solution:
We are given:
\[
\lim_{x \to \infty} \frac{\ln\left(1 + \frac{9}{x}\right)}{\frac{1}{5x}}
\]
After applying L'Hôpital's Rule once, we need to differentiate the numerator and the denominator with respect to \(x\).

#### 1. Differentiate the Numerator:
The numerator is:
\[
\ln\left(1 + \frac{9}{x}\right)
\]
Using the chain rule, the derivative is:
\[
\frac{1}{1 + \frac{9}{x}} \cdot \left(-\frac{9}{x^2}\right) = \frac{-\frac{9}{x^2}}{1 + \frac{9}{x}}
\]

#### 2. Differentiate the Denominator:
The denominator is:
\[
\frac{1}{5x}
\]
Differentiating with respect to \(x\) gives:
\[
-\frac{1}{5x^2}
\]

#### 3. Substitute into the Limit:
Now the limit becomes:
\[
\lim_{x \to \infty} \frac{\frac{-\frac{9}{x^2}}{1 + \frac{9}{x}}}{-\frac{1}{5x^2}}
\]
This simplifies to:
\[
\lim_{x \to \infty} \frac{9}{1 + \frac{9}{x}} \cdot 5 = \lim_{x \to \infty} \frac{45}{1 + \frac{9}{x}}
\]
As \(x \to \infty\), \(\frac{9}{x} \to 0\), so we get:
\[
\frac{45}{1} = 45
\]

### Final Answer:
\[
\lim_{x \to \infty} \frac{\ln\left(1 + \frac{9}{x}\right)}{\frac{1}{5x}} = 45
\]

Let me know if you'd like further details, or have any questions!

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### Follow-Up Questions
1. What is L'Hôpital's Rule and why is it used here?
2. How would the solution change if the \(9\) in \(\frac{9}{x}\) were replaced by another constant?
3. How do we interpret the result in terms of the original function \(y\)?
4. Can you apply L'Hôpital's Rule more than once in cases where it’s necessary?
5. How do you evaluate limits involving logarithmic functions as \(x \to \infty\)?

### Tip
When using L'Hôpital's Rule, ensure that the limit is indeed an indeterminate form (like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)) before applying it.

Consider the limit of ln(y) as x approaches infinity, using L'Hôpital's Rule.

Explore how to find the limit of ln(y) as x approaches infinity using L'Hôpital's Rule. This solution includes detailed steps in applying L'Hôpital's Rule to an indeterminate form involving logarithmic functions and infinity. Ideal for students in undergraduate calculus courses.

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