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The image provides a solution process that uses L'Hôpital's Rule to evaluate the limit

\[
\lim_{x \to \infty} \frac{\ln \left(1 + \frac{5}{x} \right)}{\frac{1}{5x}}.
\]

The solution applies L'Hôpital's Rule, which requires taking the derivatives of the numerator and the denominator separately.

1. **Differentiate the Numerator**:
   \[
   \text{Numerator: } \ln \left( 1 + \frac{5}{x} \right).
   \]
   Its derivative is:
   \[
   \frac{1}{1 + \frac{5}{x}} \cdot \left(-\frac{5}{x^2}\right) = -\frac{5}{x^2 \left(1 + \frac{5}{x}\right)}.
   \]

2. **Differentiate the Denominator**:
   \[
   \text{Denominator: } \frac{1}{5x}.
   \]
   Its derivative is:
   \[
   -\frac{1}{5x^2}.
   \]

3. **Simplify the Limit**:
   Substituting these derivatives into the limit gives:
   \[
   \lim_{x \to \infty} \frac{-\frac{5}{x^2 \left(1 + \frac{5}{x}\right)}}{-\frac{1}{5x^2}} = \lim_{x \to \infty} \frac{5}{\frac{1}{1 + \frac{5}{x}}}.
   \]
   As \( x \to \infty \), \( \frac{5}{x} \to 0 \), so \( 1 + \frac{5}{x} \to 1 \), simplifying the limit to:

   \[
   \lim_{x \to \infty} 5 = 5.
   \]

Thus, the answer in the missing box should be \( -\frac{5}{x^2} \). 

Would you like further explanation on each derivative or the application of L'Hôpital's Rule? 

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### Related Questions:
1. What is L'Hôpital's Rule, and when can it be applied?
2. How do you determine if a limit is in an indeterminate form?
3. What are the most common indeterminate forms that require L'Hôpital's Rule?
4. How does the behavior of \(\frac{1}{x}\) as \( x \to \infty \) impact limits?
5. How can we simplify expressions involving \( \frac{5}{x} \) as \( x \to \infty \)?

### Tip:
When applying L'Hôpital's Rule, always confirm that the limit is in an indeterminate form (like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)) before taking derivatives.

Evaluate the limit \( \lim_{{x \to \infty}} \frac{{\ln \left(1 + \frac{5}{x} \right)}}{{\frac{1}{5x}}} \) using L'Hôpital's Rule.

Learn to evaluate the limit \( \lim_{{x \to \infty}} \frac{{\ln \left(1 + \frac{5}{x} \right)}}{{\frac{1}{5x}}} \) using L'Hôpital's Rule. This solution involves finding derivatives and simplifying expressions to resolve an indeterminate form. Ideal for advanced high school or college-level calculus students.

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