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?

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.