Let's solve the limit:

\[
\lim_{n \to \infty} \ln \left( \frac{(n+4)^n}{n^n} \right)
\]

### Step 1: Simplify the expression inside the logarithm
We can start by breaking down the fraction inside the logarithm:

\[
\ln \left( \frac{(n+4)^n}{n^n} \right) = \ln \left( \frac{(n+4)^n}{n^n} \right) = \ln \left( (n+4)^n \right) - \ln \left( n^n \right)
\]

Using the logarithmic property \( \ln(a^b) = b \ln(a) \), we can simplify:

\[
n \ln(n+4) - n \ln(n)
\]

### Step 2: Factor out \(n\)
Now, we have:

\[
n \left( \ln(n+4) - \ln(n) \right)
\]

Using the logarithmic identity \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \), we can simplify further:

\[
n \ln\left( \frac{n+4}{n} \right)
\]

Simplifying the fraction inside the logarithm:

\[
\frac{n+4}{n} = 1 + \frac{4}{n}
\]

So, the expression becomes:

\[
n \ln \left( 1 + \frac{4}{n} \right)
\]

### Step 3: Apply limit as \( n \to \infty \)
We now evaluate the limit of \( n \ln \left( 1 + \frac{4}{n} \right) \) as \( n \to \infty \).

For large \(n\), we can use the approximation \( \ln(1+x) \approx x \) when \(x\) is small. In this case, \( x = \frac{4}{n} \) becomes very small as \( n \) approaches infinity. So, we approximate:

\[
\ln \left( 1 + \frac{4}{n} \right) \approx \frac{4}{n}
\]

Substituting this back into the expression:

\[
n \cdot \frac{4}{n} = 4
\]

### Final Answer:
Therefore, the limit is:

\[
\lim_{n \to \infty} \ln \left( \frac{(n+4)^n}{n^n} \right) = 4
\]

---

Would you like further clarification or additional details? Here are some follow-up questions to consider:

1. Why is \( \ln(1 + x) \approx x \) a good approximation for small \(x\)?
2. What is the significance of logarithmic limits in large \(n\) analysis?
3. How would the result change if the constant "4" were replaced by another value?
4. Can you think of a similar limit problem involving different functions?
5. What happens if you evaluate the limit without logarithmic properties?

**Tip:** When taking limits of logarithmic functions, applying small approximations like \( \ln(1 + x) \approx x \) for small \(x\) is often useful.

Explore the detailed solution to the limit of ln(((n+4)^n)/(n^n)) as n tends to infinity. This problem delves into key calculus concepts, such as logarithmic properties and asymptotic behavior, with a step-by-step breakdown suitable for students in advanced calculus.

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