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To solve the given limit problem:

\[
\lim_{n \to \infty} \left(1 + \frac{1}{3n-1}\right)^{4n+3},
\]

we will proceed using the following steps:

### Step 1: Rewrite the expression
Notice that the expression resembles the form of \( \left(1 + \frac{1}{x}\right)^x \), which approaches \( e \) as \( x \to \infty \). We will try to rewrite the given limit to use this fact.

Let’s rewrite \( \frac{1}{3n-1} \) as \( \frac{1}{3n(1 - \frac{1}{3n})} \approx \frac{1}{3n} \) for large \( n \), since \( \frac{1}{3n} \to 0 \).

### Step 2: Simplify the exponent
The original problem becomes approximately:

\[
\lim_{n \to \infty} \left(1 + \frac{1}{3n}\right)^{4n+3}.
\]

### Step 3: Factor the exponent
Rewrite the exponent \( 4n + 3 \) as \( 3n \left( \frac{4n+3}{3n} \right) \), giving:

\[
\lim_{n \to \infty} \left(1 + \frac{1}{3n}\right)^{3n \cdot \frac{4n+3}{3n}}.
\]

### Step 4: Apply the limit properties
As \( n \to \infty \), \( \frac{4n+3}{3n} \to \frac{4}{3} \), so the expression becomes:

\[
\left[\left(1 + \frac{1}{3n}\right)^{3n}\right]^{\frac{4}{3}}.
\]

### Step 5: Use the known limit
The limit \( \lim_{n \to \infty} \left(1 + \frac{1}{3n}\right)^{3n} = e \). Therefore, the original expression becomes:

\[
e^{\frac{4}{3}}.
\]

Thus, the final answer is:

\[
\lim_{n \to \infty} \left(1 + \frac{1}{3n-1}\right)^{4n+3} = e^{\frac{4}{3}}.
\]

Would you like a more detailed explanation of any of these steps? Here are five related questions for further exploration:

1. How does the form \( \left(1 + \frac{1}{n}\right)^n \to e \) as \( n \to \infty \)?
2. What happens if we change the base expression to \( 1 + \frac{k}{n} \), where \( k \) is a constant?
3. How do we use the approximation \( \frac{1}{3n-1} \approx \frac{1}{3n} \) for large \( n \)?
4. Why does rewriting the exponent as \( 3n \cdot \frac{4n+3}{3n} \) help solve the limit?
5. What other forms of limits result in expressions involving the number \( e \)?

### Tip:
When dealing with limits of the form \( \left(1 + \frac{1}{n}\right)^n \), try rewriting the expression to fit known limit theorems involving the number \( e \).

Evaluate the limit: \( \lim_{n \to \infty} \left(1 + \frac{1}{3n-1}\right)^{4n+3} \).

Learn how to solve the limit problem \( \lim_{n \to \infty} \left(1 + \frac{1}{3n-1}\right)^{4n+3} \). This solution involves concepts from limits and exponential functions, using properties of sequences approaching \( e \). Suitable for college-level students.

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