https://file.aiquickdraw.com/mathbot/file/17303406883854jqvx3p8.jpg

To solve this, let’s go through the steps as follows:

1. We start with \( y = \left(1 + \frac{5}{x}\right)^{5x} \).
2. Taking the natural logarithm on both sides gives:
   \[
   \ln(y) = \ln\left(\left(1 + \frac{5}{x}\right)^{5x}\right)
   \]
3. Using the logarithmic power rule \( \ln(a^b) = b \ln(a) \), we get:
   \[
   \ln(y) = 5x \cdot \ln\left(1 + \frac{5}{x}\right)
   \]

So, the answer to fill in the blank is:
\[
5x \cdot \ln\left(1 + \frac{5}{x}\right)
\]

Let me know if you'd like further details on the steps or have additional questions.

Here are some related questions:
1. Why do we use logarithms to solve exponential limits?
2. How does this approach help with limits that involve expressions of the form \( (1 + \frac{k}{n})^n \)?
3. What role does L'Hôpital's Rule play in evaluating limits of sequences?
4. Can this type of limit be solved using Taylor expansion?
5. How do we determine when to use natural logarithms in limit problems?

**Tip:** When faced with expressions of the form \( (1 + \frac{k}{n})^n \), logarithmic transformation often simplifies the limit by allowing the exponent to be dealt with separately.

The given limit of \( a_n = \left( 1 + \frac{5}{n} \right)^{5n} \) cannot be evaluated directly. We will first find the limit of a related sequence. Let \( y = \left( 1 + \frac{5}{x} \right)^{5x} \). Then taking the natural logarithm of each side of the equation results in the following: \( \ln(y) = \ln\left( \left( 1 + \frac{5}{x} \right)^{5x} \right) = ? \).

Learn how to solve the exponential limit \( \left( 1 + \frac{5}{n} \right)^{5n} \) by using logarithmic transformation. This problem covers key calculus concepts, including limits, exponential growth, and the natural logarithm function. Suitable for undergraduate calculus students.

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